# 19: Equations and Inequalities

In this chapter, you will explore linear equations, develop a strategy for solving them, and relate them to real-world situations.

## US racial inequality may be as deadly as COVID-19

The COVID-19 pandemic is causing a catastrophic increase in US mortality. How does the scale of this pandemic compare to another US catastrophe: racial inequality? Using demographic models, I estimate how many excess White deaths would raise US White mortality to the best-ever (lowest) US Black level under alternative, plausible assumptions about the age patterning of excess mortality in 2020. I find that 400,000 excess White deaths would be needed to equal the best mortality ever recorded among Blacks. For White mortality in 2020 to reach levels that Blacks experience outside of pandemics, current COVID-19 mortality levels would need to increase by a factor of nearly 6. Moreover, White life expectancy in 2020 will remain higher than Black life expectancy has ever been unless nearly 700,000 excess White deaths occur. Even amid COVID-19, US White mortality is likely to be less than what US Blacks have experienced every year. I argue that, if Black disadvantage operates every year on the scale of Whites’ experience of COVID-19, then so too should the tools we deploy to fight it. Our imagination should not be limited by how accustomed the United States is to profound racial inequality.

The COVID-19 pandemic is likely to kill people in the United States on a scale not seen in a century, since the 1918 flu. That catastrophic flu killed tens of millions worldwide and over half a million in the United States. Yet mortality levels that, for White Americans, were unprecedented nevertheless were lower than mortality for US Blacks in any given (nonpandemic) year. Past research has shown that the infectious mortality experienced by urban Whites in 1918 was lower than the infectious mortality of urban nonwhites in every documented year through 1920 (1). Fig. 1 shows that a similar pattern pertains to Whites’ and Blacks’ age-adjusted total mortality and life expectancy. Whites’ life expectancy in 1918 was far lower than in other twentieth century years—yet higher than Black life expectancy in all years but one between 1900 and 1918. Similarly, Whites’ age-adjusted mortality in 1918 was lower than Black mortality in all years but one from 1900 to 1931. By all major mortality measures—infectious mortality, total mortality, and life expectancy—in the early decades of the twentieth century, Blacks in the United States experienced a scale of death comparable to Whites’ experience of the 1918 flu every year.

US Black and White (A) logged deaths per 100,000 and (B) life expectancy, 1900–2017.

A century later, stark inequalities in survival persist. Will graphs of the early twenty-first century look like graphs of the early twentieth century, with a deadly pandemic causing a spike in mortality for Whites that nevertheless remains lower than the mortality Blacks experience routinely, outside of any pandemic? This question cannot be answered definitively until the final toll of COVID-19 is known. As a framework for answering it, I estimate how many White deaths from COVID-19 would be required for White mortality in 2020 to reach the levels of Black mortality in its best recorded year.

The results provide context for understanding the scale of racial inequality in mortality in the United States. Despite recent scholarly focus on rising White mortality (2), that racial inequality remains extreme. As Fig. 1 underscores, best-ever Black age-adjusted mortality and life expectancy are equivalent to White rates from, respectively, nearly 20 or 30 y earlier. For COVID-19 to raise mortality as much as racial inequality does, it would need to erase two to three decades of mortality progress for Whites.

## Lesson 19

A teacher is choosing between two options for a class field trip to an orchard.

• At Orchard A, admission costs $9 per person and 3 chaperones are required. • At Orchard B, the cost is$ 10 per person, but only 1 chaperone is required.
• At each orchard, the same price applies to both chaperones and students.  Expand Image Attribution: Apple tree, by pxhere. Public Domain. pxhere. Source.

Which orchard would be cheaper to visit if the class has:

To help her compare the cost of her two options, the teacher first writes the equation (9(n + 3) = 10(n + 1)) , and then she writes the inequality (9(n + 3) < 10(n + 1)) .

1. What does (n) represent in each statement?
2. In this situation, what does the equation (9(n + 3) = 10(n + 1)) mean?
3. What does the solution to the inequality (9(n + 3) < 10(n + 1)) tell us?

Graph the solution to the inequality on the number line. Be prepared to show or explain your reasoning. Expand Image

### 19.3: Part-Time Work

To help pay for his tuition, a college student plans to work in the evenings and on weekends. He has been offered two part-time jobs: working in the guest-services department at a hotel and waiting tables at a popular restaurant.

• The job at the hotel pays $18 an hour and offers$ 33 in transportation allowance per month.
• The job at the restaurant pays $7.50 an hour plus tips. The entire waitstaff typically collects about$ 50 in tips each hour. Tips are divided equally among the 4 waitstaff members who share a shift.

The equation (7.50h + frac <50><4>h = 18h + 33) represents a possible constraint about a situation.

1. Solve the equation and check your solution.
2. Here is a graph on a number line. Expand Image

Put a scale on the number line so that the point marked with a circle represents the solution to the equation.

Does one job pay better if:

1. The student works fewer hours than the solution you found earlier? If so, which job?
2. The student works more hours than the solution you found earlier? If so, which job?

Be prepared to explain or show how you know.

Here are two inequalities and two graphs that represent the solutions to the inequalities. Expand Image Expand Image

1. Put the same scale on each number line so that the circle represents the number of hours that you found earlier.
2. Match each inequality with a graph that shows its solution. Be prepared to explain or show how you know.

### 19.4: Equality and Inequality

1. Solve this equation and check your solution: (displaystyle ext-frac<4(x+3)>5 = 4x-12) .
2. Consider the inequality: (displaystyle ext-frac<4(x+3)>5 le 4x-12) .
1. Choose a couple of values less than 2 for (x) . Are they solutions to the inequality?
2. Choose a couple of values greater than 2 for (x) . Are they solutions to the inequality?
3. Choose 2 for (x) . Is it a solution?

Graph the solution to the inequality on the number line. Expand Image

Here is a different type of inequality: (x^2 leq 4) .

1. Is 1 a solution to the inequality? Is 3 a solution? How about -3?
2. Describe all solutions to this inequality. (If you like, you can graph the solutions on a number line.)
3. Describe all solutions to the inequality (x^2 geq 9) . Test several numbers to make sure your answer is correct.

### 19.5: More or Less?

Consider the inequality ( ext <->frac12 x + 6 < 4x−3) . Let's look at another way to find its solutions.

Use the provided applet or another graphing technology to graph (y= ext <->frac12 x + 6 ) and (y=4x−3) on the same coordinate plane.

Use your graphs to answer the following questions. If using the applet, the slider might be helpful.

1. Find the values of ( ext-frac <1><2>x+6) and (4x-3) when (x) is 1.
2. What value of (x) makes ( ext-frac <1><2>x+6) and (4x-3) equal?
3. For what values of (x) is ( ext- frac12 x + 6) less than (4x−3) ?
4. For what values of (x) is ( ext- frac12 x + 6) greater than (4x−3) ?

### Summary

The equation (frac12 t = 10) is an equation in one variable. Its solution is any value of (t) that makes the equation true. Only (t=20) meets that requirement, so 20 is the only solution.

The inequality (frac12t >10) is an inequality in one variable. Any value of (t) that makes the inequality true is a solution. For instance, 30 and 48 are both solutions because substituting these values for (t) produces true inequalities. (frac12(30) >10) is true, as is (frac12(48) >10) . Because the inequality has a range of values that make it true, we sometimes refer to all the solutions as the solution set.

One way to find the solutions to an inequality is by reasoning. For example, to find the solution to (2p<8) , we can reason that if 2 times a value is less than 8, then that value must be less than 4. So a solution to (2p<8) is any value of (p) that is less than 4.

Another way to find the solutions to (2p<8) is to solve the related equation (2p=8) . In this case, dividing each side of the equation by 2 gives (p=4) . This point, where (p) is 4, is the boundary of the solution to the inequality.

To find out the range of values that make the inequality true, we can try values less than and greater than 4 in our inequality and see which ones make a true statement.

Let's try some values less than 4:

• If (p=3) , the inequality is (2(3) <8) or (6 < 8) , which is true.
• If (p= ext-1) , the inequality is (2( ext-1) < 8) or ( ext-2 <8 ) , which is also true.

Let's try values greater than 4:

• If (p=5) , the inequality is (2(5)<8) or (10<8) , which is false.
• If (p=12) , the inequality is (2(12) <8) or (24<8) , which is also false.

In general, the inequality is false when (p) is greater than or equal to 4 and true when (p) is less than 4.

We can represent the solution set to an inequality by writing an inequality, (p<4) , or by graphing on a number line. The ray pointing to the left represents all values less than 4. Expand Image

## 19.2: Off to an Orchard (20 minutes)

### Activity

This activity encourages students to interpret an inequality and its solution set in terms of a situation. A context can help students intuit why the solutions to an inequality form a ray on the number line.

The activity also prompts students to think about the solutions to an inequality in terms of a related equation. Here the situation involves choosing between two options. An equation can be written to represent the two options being equal. The solution to that equation can be seen as a tipping point, on either side of which one option would be better.

Students may recall this way of solving inequalities from middle school, but they may also solve by testing different possible values, or by reasoning about the relationship between quantities in other ways. The work here encourages students to reason quantitatively and abstractly (MP2), and to make sense of problems and persevere in solving them (MP1).

Monitor the strategies students use to find the solutions to (9(n + 3) < 10(n + 1)) , and identify students using different approaches. Students may:

• Try different values of (n) until the inequality is no longer true.
• Try different numbers higher than 12 (based on their work on the first question) and find that, up to 17 students, the cost to go to Orchard B is lower. Beyond 17 students, the cost for Orchard A is lower.
• Solve the equation (9(n + 3) = 10(n + 1)) to find the number of students at which the costs for both options would be equal. That number is 17. Then, try a higher or lower number to see which side of the equation has a smaller value.
• Reason about the difference in the cost per student and cost for chaperones. The cost per student at Orchard A ( $9) is$ 1 lower than at Orchard B ( $10). But because 3 chaperones are required at Orchard A ($ 27 for 3 chaperones) and only 1 at Orchard B ( $10 for 1 chaperone), the cost for chaperones is$ 17 higher at Orchard A than at Orchard B. So if 17 students go on the trip, the cost would be the same at both places. If more than 17 students go, Orchard A would be cheaper.

Some students may find the solutions to (9(n + 3) < 10(n + 1)) by manipulating the inequality to isolate (n) . Depending on the operations performed, they may once again end up with an incorrect solution set if they forget to reverse the inequality symbol. (For example, in the final step of solving, they may go from ( ext-1n < ext-17) to (n <17) .) If this happens, bring the issue to students' attention during activity synthesis.

### Launch

Read the first part of the task statement with the class and make sure students understand the given information.

Arrange students in groups of 2. For the first set of questions, ask one partner to find the cost of going to Orchard A and the other partner to find the cost of going to Orchard B, and then compare the costs. Before students move on to the second set of questions, pause to hear which option works best for 8, 12, and 30 students.

A teacher is choosing between two options for a class field trip to an orchard.

• At Orchard A, admission costs $9 per person and 3 chaperones are required. • At Orchard B, the cost is$ 10 per person, but only 1 chaperone is required.
• At each orchard, the same price applies to both chaperones and students.  Expand Image Attribution: Apple tree, by pxhere. Public Domain. pxhere. Source.

Which orchard would be cheaper to visit if the class has:

To help her compare the cost of her two options, the teacher first writes the equation (9(n + 3) = 10(n + 1)) , and then she writes the inequality (9(n + 3) < 10(n + 1)) .

1. What does (n) represent in each statement?
2. In this situation, what does the equation (9(n + 3) = 10(n + 1)) mean?
3. What does the solution to the inequality (9(n + 3) < 10(n + 1)) tell us?

Graph the solution to the inequality on the number line. Be prepared to show or explain your reasoning. Expand Image

## Example 3: Variables on Both Sides and Dividing By a Negative Number

Let's take a look at how our new rule applies to solving equations. Pay special attention to the part where we divide by negative 1. Our last example revisits how to solve equations and/or inequalities with fractions. I hope you remember the trick.

## 19: Equations and Inequalities

More than ever, students need to engage with mathematical concepts, think quantitatively and analytically, and communicate using mathematics. All these skills are central to a young person’s preparedness to tackle problems that arise at work and in life beyond the classroom. But the reality is that many students are not familiar with basic mathematics concepts and, at school, only practice routine tasks that do not improve their ability to think quantitatively and solve real-life, complex problems.

How can we break this pattern? This report, based on results from PISA 2012, shows that one way forward is to ensure that all students spend more “engaged” time learning core mathematics concepts and solving challenging mathematics tasks. The opportunity to learn mathematics content – the time students spend learning mathematics topics and practising maths tasks at school – can accurately predict mathematics literacy. Differences in students’ familiarity with mathematics concepts explain a substantial share of performance disparities in PISA between socio-economically advantaged and disadvantaged students. Widening access to mathematics content can raise average levels of achievement and, at the same time, reduce inequalities in education and in society at large.

In Unit 6, sixth graders move from expressions to equations and inequalities. They revisit familiar diagrams such as tape diagrams to model equations, and they discover new models such as balances and hanging mobiles. Students investigate what it means to be a solution to an equation or an inequality and how use equations and inequalities to model relationships between quantities. When using an equation or inequality to represent real-world situations, students must decontextualize the situation to represent it using variables and symbols and then recontextualize in order to interpret what their answer means in regard to the situation at hand (MP.2). In this unit, students bring concepts from three domains together: Ratios and Proportions, Number Sense, and Expressions and Equations. They re-visit percentages from Unit 2 and solve percent problems using equations. They study relationships between different quantities and draw on their ratio reasoning where relevant. A note on fluency: solving equations provides a good opportunity for students to continue development of and to demonstrate fluency with decimal operations and fraction division. Several problems involve computing with decimals and dividing by fractions include additional problems in practice for students as needed.

Several prior skills support students in this unit. In fifth grade, students analyzed patterns and relationships when they studied standard 5.OA.3. They also observed what happened when these relationships were plotted on the coordinate plane. In previous sixth-grade units, students studied algebraic and numerical expressions and collections of equivalent ratios. Students draw on all of these concepts and skills in this unit.

There are many future connections to the standards in this unit. In seventh grade, students will deeply investigate proportional relationships in the form $y=rx$ , understanding the value of $r$ as the constant of proportionality. They&rsquoll further investigate the graphs of these equations, and in eighth grade, students will compare across multiple representations of proportional relationships. Students will also become exposed to increasingly more complex equations and inequalities to solve.

Pacing: 17 instructional days (14 lessons, 2 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 6th Grade Scope and Sequence Recommended Adjustments.

## # References

Adams-Prassl, A., Boneva, T., Golin, M., & Rauh, C. (2020a). Inequality in the Impact of the Coronavirus Shock: Evidence from Real Time Surveys. Journal of Public Economics.

Adams-Prassl, A., Boneva, T., Golin, M., & Rauh, C. (2020b). Work Tasks that Can Be Done from Home: Evidence on Variation within & Across Occupations and Industries. IZA Discussion Paper 13374.

Alon, T. M., Doepke, M., Olmstead-Rumsey, J., & Tertilt, M. (2020). The impact of COVID-19 on gender equality (No. w26947). National Bureau of Economic Research.

Andrew, A., Cattan, S., Costa Dias, M., Farquharson, C., Kraftman, L., Krutikova, S., . & Sevilla, A. (2020). The gendered division of paid and domestic work under lockdown. SSRN Papers.

Angrave, D., & Charlwood, A. (2015). What is the relationship between long working hours, over-employment, under-employment and the subjective well-being of workers? Longitudinal evidence from the UK. Human Relations, 68(9), 1491-1515.

Benzeval, M., Burton, J., Crossley, T. F., Fisher, P., Jäckle, A., Low, H., & Read, B. (2020). The Idiosyncratic Impact of an Aggregate Shock: The Distributional Consequences of COVID-19. Available at SSRN 3615691.

Bick, A., & Blandin, A. (2020). Real-time labor market estimates during the 2020 coronavirus outbreak. Available at SSRN 3692425.

Bick, A., Blandin, A., & Mertens, K. (2020). Work from home after the COVID-19 Outbreak. SSRN Papers.

Blanchflower, D. G., & Oswald, A. J. (2008). Is well-being U-shaped over the life cycle?. Social science & medicine, 66(8), 1733-1749.

Bloom, N., Liang, J., Roberts, J., & Ying, Z. J. (2015). Does working from home work? Evidence from a Chinese experiment. The Quarterly Journal of Economics, 130(1), 165-218.

Blundell, R., Costa Dias, M., Joyce, R., & Xu, X. (2020). COVID‐19 and Inequalities. Fiscal Studies, 41(2), 291-319.

Brewer, M., Gardiner, L., & Handscomb, K. (2020). The truth will out: understanding labour market statistics during the coronavirus crisis: July 2020.

Carlson, D. L., Petts, R., & Pepin, J. (2020). US Couples' Divisions of Housework and Childcare During COVID-19 Pandemic.

Chetty, R., Friedman, J., Hendren, N., & Stepner, M. (2020). The economic impacts of COVID-19: Evidence from a new public database built from private sector data. Opportunity Insights. Retrieved from: www.tracktherecovery.org

Clark, A. E. (2003). Unemployment as a social norm: Psychological evidence from panel data. Journal of Labor Economics, 21(2), 323-351.

Clark, A. E., Flèche, S., Layard, R., Powdthavee, N., & Ward, G. (2019). The origins of happiness: the science of well-being over the life course. Princeton University Press.

Coibion, O., Gorodnichenko, Y., & Weber, M. (2020). Labor markets during the covid-19 crisis: A preliminary view (No. w27017). National Bureau of Economic Research.

Collins, C., Landivar, L. C., Ruppanner, L., & Scarborough, W. J. (2020). COVID‐19 and the gender gap in work hours. Gender, Work & Organization.

Cotofan, M., Cassar, L., Dur, R., & Meier, S. (2020). Macroeconomic Conditions When Young Shape Job Preferences for Life.

Czeisler et al. (2020). Mental health, substance use, and suicidal ideation during the COVID-19 pandemic---United States, June 24--30, 2020. Morbidity and Mortality Weekly Report, 69 (32), 1049.

Davis, M. A., Ghent, A. C., & Gregory, J. (2021). The Work-at-Home Technology Boon and its Consequences. Available at SSRN 3768847.

De Neve, J. E., & Oswald, A. J. (2012). Estimating the influence of life satisfaction and positive affect on later income using sibling fixed effects. Proceedings of the National Academy of Sciences, 109(49), 19953-19958.

De Neve, J. E., & Ward, G. (2017). Happiness at work. In World Happiness Report 2017.

Del Boca, D., Oggero, N., Profeta, P., & Rossi, M. (2020). Women's Work, Housework and Childcare, before and during COVID-19.

Dingel, J. I., & Neiman, B. (2020). How many jobs can be done at home? (No. w26948). National Bureau of Economic Research.

Dolan, P., Peasgood, T., & White, M. (2008). Do we really know what makes us happy? A review of the economic literature on the factors associated with subjective well-being. Journal of economic psychology, 29(1), 94-122.

Eurostat (2020). COVID-19 labour effects across the income distribution. European Commission. Retrieved from: https://ec.europa.eu/eurostat/web/products-eurostat-news/-/DDN-20201027-2

Foa, R., Gilbert, S., & Fabian, M. O. (2020). COVID-19 and subjective well-being: Separating the effects of lockdowns from the pandemic. Available at SSRN 3674080.

Galasso, V. (2020). Covid: not a great equaliser. Covid Economics Vetted and Real-Time Papers 1(19): 241-265.

Giurge, L. M., Macchia, L., Whillans, A. V., & Yemiscigil, A. (forthcoming). How COVID-19 Shapes Students' Extrinsic and Prosocial Work Values: The Role of Uncertainty, Worry and Reflection.

Gottlieb, C., Grobovšek, J., & Poschke, M. (2020). Working from home across countries. COVID Economics, 1(8): 71-91.

Happiness Research Institute (2020) Well-being in the age of COVID-19, Copenhagen: Happiness Research Institute.

Hatayama, M., Viollaz, M., & Winkler, H. (2020). Jobs' Amenability to Working from Home: Evidence from Skills Surveys for 53 Countries. COVID Economics, 1(19): 211-240.

Helliwell, J., Schellenberg, G., and Fonberg, J. (2020). Life satisfaction in Canada before and during the COVID-19 pandemic. Ottawa: Statistics Canada Analytical Branch Research Paper Series https://www150.statcan.gc.ca/n1/pub/11f0019m/11f0019m2020020-eng.htm

Hertz, R., Mattes, J., & Shook, A. (2020). When Paid Work Invades the Family: Single Mothers in the COVID-19 Pandemic. Journal of Family Issues, 0192513X20961420.

Hughes, M. E., Waite, L. J., Hawkley, L. C., & Cacioppo, J. T. (2004). A short scale for measuring loneliness in large surveys: Results from two population-based studies. Research on aging, 26(6), 655-672.

Hupkau, C., & Petrongolo, B. (2020). Work, care and gender during the Covid-19 crisis. CEP Discussion Paper, No. 1723. Centre for Economic Performance: London.

Indeed Hiring Lab (2020). Indeed Hiring Lab Data [database]. Available from: https://github.com/hiring-lab/data

International Labour Organization (2020a). ILO Monitor: COVID-19 and the world of work. Sixth edition. Switzerland: ILO.

International Labour Organization (2020b). ILO Monitor: COVID-19 and the world of work. Third edition. Switzerland: ILO.

International Labour Organization (2020c). ILO Monitor: COVID-19 and the world of work. Fifth edition. Switzerland: ILO.

International Labour Organization (2020d). ILO Monitor: COVID-19 and the world of work. Fourth edition. Switzerland: ILO.

International Labour Organization. (2020e). ILOSTAT database [database]. Available from: www.ilostat.ilo.org/data

International Labour Organization (2021). ILO Monitor: COVID-19 and the world of work. Seventh edition. Switzerland: ILO.

International Monetary Fund (2020). World Economic Outlook: A Long and Difficult Ascent. Washington, DC: IMF.

Imperial College London Big Data Analytical Unit and YouGov (2020). Imperial College London YouGov Covid Data Hub, v1.0 [database]. Available from: https://github.com/YouGov-Data/covid-19-tracker

Kikuchi, S., Kitao, S., & Mikoshiba, M. (2020). Who Suffers from the COVID-19 Shocks? Labor Market Heterogeneity and Welfare Consequences in Japan. Covid Economics, 1(40): 76-114.

Kramer, S. (2019). U.S. has world's highest rate of children living in single-parent households. Pew Research Center. Retrieved from: https://pewrsr.ch/2LLvbxW

Krekel, C., Ward, G., & De Neve, J. E. (2019). What makes for a good job? Evidence using subjective well-being data. In The Economics of Happiness (pp. 241-268). Springer, Cham.

Kreyenfeld, M., & Zinn, S. (2020). Coronavirus & care: How the coronavirus crisis affected fathers' involvement in Germany (No. 1096). SOEPpapers on Multidisciplinary Panel Data Research.

Lebowitz, S. (2020). You could save more than \$10,000 a month by getting rid of your company's offices. 5 CEOs who did just that told us how they made the decision. Business Insider. Retrieved from: https://www.businessinsider.com/ceos-no-offices-fully-remote-virtual-work-coronavirus-pandemic-recession-2020-10

Lee, I., & Tipoe, E. (2020). Time use and productivity during the COVID-19 lockdown: Evidence from the UK. IZA Working Paper.

Marinescu, I. E., Skandalis, D., & Zhao, D. (2020). Job search, job posting and unemployment insurance during the COVID-19 crisis. Job Posting and Unemployment Insurance During the COVID-19 Crisis (July 30, 2020).

Mertens, K., Blandin, A., & Bick, A. (2020). Work from Home After the COVID-19 Outbreak.

Mongey, S., Pilossoph, L. & Weinberg, A. (2020). Which workers bear the burden of social distancing policies? (No. w27085). National Bureau of Economic Research.

Morikawa, M. (2020). Productivity of Working from Home during the COVID-19 Pandemic: Evidence from an Employee Survey (Japanese). Research Institute of Economy, Trade and Industry (RIETI).

OECD (2020a). Job retention schemes during the COVID-19 lockdown and beyond. Paris: Organization for Economic Co-operation and Development.

OECD (2020b). OECD Employment Outlook 2020: Worker Security and the COVID-19 Crisis. OECD Publishing, Paris. DOI: https://www.oecd-ilibrary.org/employment/oecd-employment-outlook-2020_1686c758-en

OECD (2021), Employment rate (indicator). DOI: 10.1787/1de68a9b-en (Accessed January 2021).

Pilipiec, P., Groot, W., & Pavlova, M. (2020). A longitudinal analysis of job satisfaction during a recession in The Netherlands. Social Indicators Research, 149(1), 239-269.

Rinne, U., & Zimmermann, K. F. (2012). Another economic miracle? The German labor market and the Great Recession. IZA Journal of Labor Policy, 1(1), 3.

Sevilla, A., & Smith, S. (2020). Baby steps: The gender division of childcare during the COVID-19 pandemic. COVID Economics, 1(23): 58-78.

Shafer, K., Milkie, M., & Scheibling, C. (2020). The Division of Labour Before & During the COVID-19 Pandemic in Canada.

University College London (2020). Covid-19 Social Study [database]. Private access data. For more information, visit: www.covidsocialstudy.org

Van der Meer, P. H. (2014). Gender, unemployment and subjective well-being: Why being unemployed is worse for men than for women. Social Indicators Research, 115(1), 23-44.

World Bank (2020). Global Economic Prospects, June 2020. Washington, DC: World Bank. DOI: 10.1596/978-1-4648-1553-9.

World Bank (2021). Global Economic Prospects, January 2021. Washington, DC: World Bank. DOI: 10.1596/978-1-4648-1612-3.

Zhou, M., Hertog, E., Kolpashnikova, K., & Kan, M. Y. (2020). Gender inequalities: Changes in income, time use and well-being before and during the UK COVID-19 lockdown.

Zhou, Y., Zou, M., Woods, S. A., & Wu, C. H. (2019). The restorative effect of work after unemployment: An intraindividual analysis of subjective well-being recovery through reemployment. Journal of applied psychology, 104(9), 1195.

## 19: Equations and Inequalities

Bing users found us today by entering these keywords :

• how to do linear system by combination
• Trigonometry sample question paper class - XI
• emath ks3 sats
• algebraically solving exponents and logarithms
• prime numbers java exercise
• ti 83 partial expansion
• solving ratio algebra problems
• TRigonometry chart
• newton-raphson solve nonlinear matlab system
• mcdougal littell algebra 2 book answers
• objective math
• sample investigatory project on algebra
• simplify equations online
• square roots with exponents
• math sheets only to print
• 9th grade math tutorials with solutions
• How Do You Change a Decimal to a Mixed Number?
• solve by system of quadratic equations calculator
• mcdougal littell math taks anwsers
• i can't work out this algebra question how can i found out how to solve it online?
• glencoe 8th grade science taks
• solving matrices simultaneous equation using graphic calculator
• algerba test
• subtracting math fractions with uneven denominators
• free trigonometry books and cds on line
• gmat Exponential revision
• activities and worksheets on Greatest Common Factor and Least Common Multiple
• science games year8
• alebra slopes and y-intercept
• glencoe algebra 2 answers worksheet
• calculator for fractions (PROPORTIONS)
• scientific radical square root calculator
• how to find the domain
• multiplying and dividing equation
• subtraction of negative numbers worksheet
• algebra tile worksheet
• worksheets ks3 coordinates
• What Is the Hardest Math Equation in the World?
• Why Was Algebra Invented
• nonlinear equation + printable worksheets
• +Mathematic sample Questions
• trinomial calculation
• what are principle of leaf area meter ppt
• how to find cube root decimal
• graph in slope-intercept worksheet
• solve square root and radical
• second order difference equations matrix
• Algebra connections volume two Answers for chapter 9 worksheet previous material
• find square root of quadratic equations
• eog 4th grade sample items
• solving complex exponential equations
• discount word problems worksheet
• intercept and slope calculator
• free online step by step instruction divide decimals
• how to solve RADICAL AND RATIONAL EXPONENTS
• free equation writer
• GED algebra example problems
• exponential expression examples
• slope worksheets
• Simplifying algebra equations
• MAth GRE practice with graphs, charts
• solving nonlinear equations in matlab
• scale+math
• printout math sheets for third grade
• worksheet: elementary algebra
• pratice taks test
• turn fractions into decimals calculator
• ratio formula
• Linear equation worksheets
• pre-algebra equation problem worksheet
• calculating partial fractions online
• easy way to learn the standard 8th grade algebra
• free ks3 science past papers
• solving nonlinear differential equations
• freee worksheets on probability
• ti-83 algebra work sheets
• finding least common multiples of equations
• b^2-4ac>0
• math homework checker
• algebra II real life use
• log problems on TI-83
• unit step functions on the ti-89
• substitution method with a square root
• ky free on line 6th grade math
• fractions - for beginners - printable worksheets
• Practice Math TAKS test for 6th grade
• algebra refresher online test
• how do you simplify a radical equation?
• monomial simplifier
• how to solve fractions
• General Aptitude Questions
• online gce o level maths help geometry
• mathematics problem solver
• printable division math papers
• Algebraic Expressions Worksheet
• sample test paper for 6th grade CA
• how do you put x in a graphic calculator?
• online nonlinear equation solver symbolically
• answers to mathematics structure and method course 2
• algebrator gaussian
• hard math equations
• free worksheets for identify and write mixed numbers for fourth grade
• Algebra 1 Worksheets
• college algebra calculator
• aptitude test paper with answers
• algebra calculator square root
• teaching radical signs to algebra students
• 11th grade math sites were you can get free worksheets
• "prentice hall mathematics course 2" "study guide" answer key
• algebra homework free
• free math worksheets for first grade odd even
• sample investigatory project in math
• free worksteets 9th grade algebra
• convert square root to decimal
• factoring expression-math
• Solve Rational Expressions
• the hardest number math problem
• convert each function from general form to standard form + worksheet
• free online tuter practice year6 sats
• online video for function mathematics grade 11
• math ks2 missing signs
• solving math questions dealing with inequalities
• printable blank coordinate planes
• solving systems of linear equations subtraction
• adding rational expressions with like denominators worksheeets
• mathematical investigatory project
• a java program to make sum of integers without of zero
• math is fun - scale factors
• ti-calc program polysmlt
• coordinate planes with pictures worksheets third grade
• math poems
• FREE PRINTABLE EOG 6TH GRADE HELP
• basic divison calculator
• graph nonlinear systems of equations substitution method
• how to simplify cubed roots with powers in them
• graphing hyperbolas on a ti-83
• definition of a radical exponent
• simplifying expressions calculator
• calculas formula
• hyperbola equation solver
• mental maths sample paper y6 free
• online ks3 practice sats papers
• accelerated math printable worksheets
• basic algebra quiz 5th grade
• solution of nonlinear differential equations
• Buisness Calculus for dummies
• math ks2 worksheets
• permutation and combination and calculator
• convert complex 5th roots
• 0,0625 base 8 base 10
• solving rational equations square roots
• online graphing calculator log quadratic
• add subtract multiply divide sums worksheet
• worksheet proportions percent
• worksheet on finding missing angles for fifth graders
• division worksheets "zeros in the quotient"
• least common denominator worksheets
• Elementary step by step approach Statistics worksheet
• maple plots in 3-d using points
• find runge kutta 3rd order method
• Steps to Solve Equations by Factoring
• Rules for adding, subtracting, multiplying, and dividing decimal numbers
• free algebra refresher course
• free coordinates worksheets
• learn how to solve system equations
• hardest math problem
• cost accounting doing the books
• science test online for 7th graders
• free dowanload of aptitude books
• cost accounting formulas
• worksheet on simplification
• word problems with algebraic expressions 4th grade
• math test papers ks2
• balance worksheets for algebra equations
• fraction formula
• evaluating and converting formulas
• solving logarithmic equations worksheet
• middle school math printouts
• solve systems of equations elimination method with square roots
• online pemdas calculator
• how to find the roots of 3rd order quadratic equation
• algebra 2 poems
• worksheet on solving equations ks3
• how to plot polynomial equation
• online maths for dummies
• combination and permutation activity middle school
• Good Aptitude question
• algebra fx 2 plus programming tutorial
• answers to california algebra 1 textbook
• dot math game - solving integers
• subtracting exponents with variable bases
• algebra substitution calculator, free
• synthetic division calculator
• TI-30XA log base
• completing the square activity
• who wants to be a mathematician elementary game
• ks2 sats science revision worksheets
• fractions of product cube
• free sheets on solving simple equations with addition
• decimal to fraction calc
• Directions for TI-85 logarithms
• graphing equalities
• parabola in algebra
• aptitude question paper of mainstay
• whole number subtraction worksheet
• Saxon math story problem worksheets for 65 textbook
• help me to understand algebra
• college algebra
• boolean algebra questions
• solving second order diffirential equations in matlab
• wallis formula and binomial
• help in prentice hall algebra 1
• math pie sign
• 0.416666667 convert to fraction
• least square trivial example slope
• linear programming calculator online
• slope of a curved line
• circle equation ti-84
• aptitude sample test paper accounts
• online algebra step by step solvers
• how to get rid of a log algebraically
• taylor TI 84 plus
• maths tests ks2
• what is the square root symbol on a calculator
• standard form second order differential equation
• algebra homework solver
• Solver for extraneous solutions
• instructions on how to calculate log2 on a TI-83
• Free programs for my TI 83
• substitution method solver
• float "E-" significant digits java convert
• extrapolation calculator
• aptitude test model paper for U G
• revised question sheet for 4th grade
• worksheets quadratic formula completing the square
• introduction to circle gcse powerpoint
• convert 1:3000 = 3%
• simultaneous solver
• convert fraction to exponent
• table of laplace transforms trigonometry
• practice taks test sheets printable answer sheet
• 10th TAKS World History quizzes and worksheets
• College Algebra Software
• purchase accounting book
• "simultaneous equations in three variables"
• symbolic method
• hardest maths equation in the world
• using graphs to determine the domain
• science sats practise printables
• adding negative and positive numbers worksheets
• Formula for Scale Factors
• math online tutor pay scale
• How can I prepare my child for the STAR 7th Grade mathematics test?
• "standard deviation inverse u shaped"
• square roots of exponents
• adding rational expressions with like denominators and worksheets
• ti calculator online graphing interactive quadratic equation parabola
• matlab combination permutation
• free printable algebra test
• standard reduction potentials movie
• vertex of linear equation
• free help with pre algebra for dummies teaching
• percent algebra formula
• simplify exponents variables
• how to solve third order equation
• double angle-trigonometric relations
• exercices et solution de matrices lineal
• fractional differenial equations
• Solving Algebra Equations
• "1st grade" and "online material"
• free online trinomial equation solver
• past maths exam papers yr8
• online maths test papers
• nonlinear graphing + printable worksheets
• ellipse lesson plan
• solving equations for kids
• lowest common multiple solver
• free 4th grade math probability worksheets
• ks2 maths graph worksheets
• Fifth Grade Solving Exponents and Division
• algebra transformation help
• 4x4 graph worksheet
• expression knocked sideways
• everyday algebra problems
• roots for cubed polynomials
• algebra 2 project
• yr 11 maths questions
• bearings free printable worksheets
• algebra 1 part 2 freeware lesson
• PRINTABLE MATH HOME WORK FOR GRADE 6TH
• equation solve mathlab
• boolean algebra simplification software
• elementary school sample circumference test
• root method
• worksheets of CPTs reading questions
• why are 6th graders disrespectful
• subtracting integers worksheet
• completing the square powepoint
• coordinate plane worksheets
• use formula to solve science TAKS
• wiki solve 3rd order equation
• where r is the base (radix) calculator
• PROBABILITY PRINTABLE WORKSHEETS KS2
• cubic function realife application
• "linear inequalities in two variables""worksheets"
• math yr8 revision
• www.softmath.com/algebra_stats2/free-online-exam-papers-on-general-yr-11-
• answers to practice 9-6 algebra 1 worksheet
• tutorials boolean algebra question with solution
• free printout assignments for my 1st grade child do do at home
• a chart of square roots of whole numbers
• free easy way to learn the 8th grade algebra
• online calculator to solve radicals
• graph relationships square root
• free college prep alegbra tutorial
• McDougal Littell Math workbook answers
• solve for binomial
• algrabra practice star test
• intercept calculator
• percentage difference equation
• cat aptitude papers
• permutation and combinations sample questions and answers
• graphing plotting points coordinate plane free worksheet
• glencoe Chemistry taks review booklet
• deriving intercept of Quadratic Equation
• free learning how to solve math equations
• math cheating machine
• elementary symmetry sheets
• tutorial on solving simultaneous equations
• Log Base 10 to Log Base 2 Conversion
• decimal to a mixed number
• 8th grade free printable worksheets
• Algebra in daily situations
• free STATS exams papers
• solving alegebra
• quadratic equation solver program for calculator
• solvoing system addition method increasing
• rationalize the numerator
• trinomial probability
• graphing ellipses and hyperbolas worksheet
• finding out a square route
• examples of math trivia with answers puzzles
• fraction multiplying calculator
• 6th grade math sat test
• explanations of ratios- fourth grade
• hyperbola focal chord
• algebra and trigonometry 2 online solutions
• free step by step algebra homework answers
• mcdougal littell pre-algebra chapter test 6 A
• a fraction to a power
• trinomial equation calculator
• mathematical formulae for compund interest
• resolving algebra
• algebric formulae
• online 9th grade algebra worksheets
• ks3 math test
• factorise algebraic hard examples
• create multiplying fractions worksheets
• high tech calcularor with cube root
• suare
• sample aptitude test questions and answers
• glencoe algebra 2 answers practice
• free online at home teaching system for 5 and 6 graders
• factoring cubed roots
• multiplication rational expressions calculator
• Polynomial Quiz Grade 8 Online
• t1-84 texas instrument calculator matrices
• 3rd order polynomial root
• convert time to decimal
• y6 mental maths sample paper
• Ontario AP Statistics Sample questions
• trigonometry cheats for problems
• formula percentage
• simultaneous linear differential equations ti-89
• coordinate plane worksheet
• "least common denominator worksheet"
• factoring and expanding quadratics worksheet
• ged expressions help
• sample essay math question for eighth grader
• women square root evil
• square root calculator online
• algebra with substitutions calculator
• hard algebra problems
• algebra homework
• variable worksheets
• square difference
• ti 83 rom image
• what is quadratic formula have to do with life
• CAT Questions on "Sequence and Series"
• what is the common FACTORS OF 20 AND 32
• Online Trinomial Factorer
• Prentice Hall Mathematics Pre-Algebra Answer Key
• create multiplying fractions worksheets
• free steps on how to solve math problems for 9th graders
• how to learn all algebra lessons very easily and free
• solved apptitude question
• maths pratice papers
• integral and rational algebraic expressions
• how to square a number on a use a ti-89
• square root to fractions
• algibra
• polynomial equation solver
• how to do laplace on ti 89
• make a sqare math game
• algebra help solving
• calculus 8th larson ebook free
• trigonometry cheat
• cube root on ti83
• pre-algebra slope homework
• Papers for teachers on the internet that are print offs for fifth graders
• calculator for calculating log
• trinomial factoring interactive game
• ising a calcultor to calculate exponent
• aptitude test question with answers
• how to solve linear transformations using ti89
• algebraic equation for kids
• integers worksheets
• online texas instruments 85 calculator demo
• pre-algebra with pizzazz!
• holt math lessons worksheets
• 3rd grade homework work sheets
• Math worksheets DD
• algebra work sheets year 8
• plotting x and y on graphing calculator
• square equation
• fraction to decimal
• practice workbook algebra 2 answers
• Solver second order ODE
• how do you get mixed numbers into a decimal
• fifth grade interacive lesson subtracting integers
• math solutions finder
• ti-86 equation substitution
• complex numbers ti 89
• linear equations worksheets gcse
• LCM exercises for fifth class
• great common divisor
• hungerford algebra exercices
• pythagoras calculator
• adding,subtracting,multiplying, and dividing integers worksheet
• matlab solve equations with summations
• sample elipse algebra 2 exercise
• adding and subtracting like denominators worksheet
• Holt Rinehart and Winston Algebra One Practice Workbook answers
• "math steps" algebra textbook
• simplify roots
• matlab decimals
• physics objective type downloadble pdf exerceses for intermediate level in India
• cubed square root in maple
• solve multiple equations on TI 89
• excel simutaneous equation solver
• Least Common Denominator Calculator
• Yr 8 Maths game
• translating exponents
• free printable taks grids
• convert word problems into expressions lesson plan
• fractions into their simplest form calculator
• algebra trigonometry problem solver free
• geometry rotation worksheets
• Calculus steps using TI-86 of solving upper and lower approximation integral problems
• apptitude test question and answers
• Free Answers To McDougal Littell textbooks
• TI-83 + circle
• Iron Works Worksheet #2 Key
• applications of simultaneous equations + grade 8+activities
• yr 11 maths
• math work sheet putting real numbers in order
• online square root calculator
• prentice hall algebra 1 tests
• intermediate maths 1A model papers
• online factorization
• free printable fraction sheets for first graders
• worksheets on integers
• algebra free worksheets transformations
• california star preparation test online for 6th grader
• algebra parabola
• 9th grade mathematics textbook online
• model aptitude test papers
• algebra for dummies online
• how to evaluate formula in javascript
• convert decimal to square root
• best algebra book
• exponential notation worksheets fourth grade
• Francois Viete substitution method for solving cubic equations
• unit math test worksheet and 5th grade
• algebra helper
• 3rd degree polynomial equation solver
• multidimensional polynomials equations solve
• square root decimal
• algebra recursive formulas free worksheet
• how to get common denominators with variables
• free printable college algebra
• square root expression
• partial fraction solver online
• ancova ls means equation
• polynomials solver
• "using roots in excel"
• synthetic division online calculator
• maple online solver
• mathematical equation to figure suare footage
• translate into equations worksheet
• Java Program to solve Algebra expression
• maths past sats papers/primary
• solve rational expressions
• factorise algebra questions
• Algebra lesson plan/graphing calulator
• ti-83 plus gaussian method
• McDougal Littell math solutions
• math problems.com
• free printable worksheets for 6th grade math problems
• easiest way to learn real analysis
• story problems for 9th grade algebra
• algebraic equations square root
• ordering positive and negative numbers
• solve a third order equation in MATLAB
• trig identities solver
• printable math pages for 3rd graders
• gcse algebra questions test
• how to solve nonhomogeneous equations
• algebra graph solvers
• algebra tutor software
• geometry formula printouts
• convert 1:3000 to a percentage
• how to teach algebraic equations
• What is the basic principle that can be used to simplify a polynomial
• Pre-Algebra fun work
• quadratc simultaneous equation solver with square
• how to solve fraction problems with pictures
• alegra problems worksheets
• nonlinear equations solution method curve cross
• SIMULTANEOUS EQUATION QUESTIONS
• Practice Sats Papers Printable
• online TI calulator
• 8th grade algebra probs for north carolina
• Converting Decimals To Fraction Calculator
• writing equations worksheets 6th grade
• rational expressions free calculator
• "free program" calculate route and distance -gps
• convert decimal to fraction calculator
• ontario math powers textbook
• interpretation of a fraction using square
• ks3 trigonometry worksheets
• binomial expansion program
• math homework problems graphing polynomial
• printable math problems.com
• Calculator tips for End of Course Algebra 2 for north carolina
• "flagg resolution"
• Adding, subtracting, multiplying and dividing rational expressions:
• 2nd grade fractions investigation lesson
• error 13 dimension
• solve quadratic equations third degree
• FIRST LESSON ON HOW TO SOLVE ALGEBRA
• tree and leaf ks3 probability
• introducing to math slope problems and answer
• multi-step equation puzzle
• maths factorise solver
• concrete volume calculater
• hyperbola CA
• multiplying percentages worksheets
• simultaneous equation +time
• free 2nd grade cummulative math test
• permutation and combinations for third graders
• help with multiplying and dividing rational expressions
• divide polynomials calculator
• free algebra calculators
• conceptual physics Teachers guide Prentice hall
• texas graphing calculator online
• percent proportion worksheets
• negative number worksheet
• first grade online probability games
• square roots of polynomials
• emath ks3 sat
• example using bellard's formula
• online equation calculator
• lowest common multiple calculator
• Long Math Poems
• Pre-Algebra Data Analysis and Probability
• prentice hall math book online
• javascript for finding the cube root of a number
• rom ti-83
• notation ti-89 quotient root
• general specific solution second order differential equation
• substitution of variables for improper integrals
• printable algebra games
• matlab convert decimal to fraction
• factor calculator
• algebra beginner
• graphing ellipse on Ti89
• prentice hall mathematics algebra 1 chapter 9 answers
• second order differential equation graph in maple
• calculate square root of inverse matrix
• how to do a quadratic equation without a graphin calculator
• printable adding subtracting integers worksheet
• Grade 10 Ontario math test
• pics of parabolas in math
• factoring polynomial.the
• multiplying integers worksheet
• how cal isolve linear eqation
• algebra websites
• Find all solutions of the equation x3 = 2 (a) in Z11
• combining like terms and constants worksheet
• factor equation online
• maths for classviii
• GRE formula list
• simplify exponential equation
• INDIANA holt science & technology SIXTH GARDE CHAPTER 16 TEST
• glencoe sample chapters world history 10th grade
• what is the decimal for 8%
• online algebra 2 book
• Printable crossnumber puzzles
• find sum of cube
• how to change de base in log operations on a ti-84
• nonlinear equation solver
• permutation and combination worksheets
• application of algebra
• solving point slope equations with fractions for dummies
• how to calculate log2 values
• algebra two vertex form practice problems
• free apptitude questions sites
• Trigonometry free sample question paper class - XI
• Sample Lesson Plan on Parabolas
• decimal to fragment
• graph the parabola worksheet x intercepts Algebra I
• math trivia for kids
• step by step easy ways for 7th graders to solve variables and expressions
• hardest easiest math problem
• free algebra for computing
• prealgerbra tests
• factoring trinomials calculator with fractions
• difference quotient from the left
• Cost Accounting Books
• percent difference equation
• test of genius answers pre-algebra with pizzazz
• CLEP College Algebra Exam Guide free
• trig power points
• free practice star test sixth grade california
• ks3 math questions
• mathematical foiling history
• examples of real life application of Euler circuit
• basic permutation combination
• permutation combination tricks
• saxon algebra 2 teacher edition
• sat ks3 algebra sheets
• median calculation fortran
• how to make hyperbolas on the ti-83 calculator
• algebra practice work sheets with pren hall
• how to factor out in algebra
• practice aptitude books
• algebraic substitution lesson plan
• binomial squaring calculator
• grade 9 Slopes worksheets math
• FORMULA PERCENTAGE OF NUMBERS
• graph systems of equations substitution method
• aptitude question on discrete mathematics
• square roots solving problems
• math, differentiation, second grade lesson plans for subtracting tens
• rudin chapter 7
• free online math solutions finder
• math for dummies
• summation integer in java
• simplify fraction matlab
• write a program in java to display number divisible by both 5 and 6.
• Pre-Algebra End Of Level Reference Sheets
• mixed number calculator
• Basic Algerbra
• i dont understand algebra
• ALGEBRA FRACTIONS WORKSHEET
• free mathematics worksheets "math worksheets" in singapore
• formula of interpolation for grading
• math statistics made incredibly easy
• circle
• i want the solution of the exercises in < basic algebra book >by jacbson
• t1-81 calculator
• what does a square root symbol look like?
• statistical three-variables possible combination calculator
• math poems on the word sum
• EOG Free Review Worksheets
• pre algerbra
• multiplying and dividing positive and negative decimals worksheets free
• second order algibra equation
• glencoe 8-7 parametric
• general solution of a second order nonhomogeneous ode
• san antonio sample tests grade placement
• rational expressions adding subtracting worksheet
• Gr.12 AP Statistics free sample questions, Ontario
• solving non linear equation excel
• mixed fraction to decimals
• rational expressions online calculator
• factors worksheet year 5
• Iowa algebra aptitude text
• quantity power algebra
• past maths exam papers yr 8
• solving cubed equations
• c# algebra fraction
• solving equations worksheets
• Rational Expressions Calculator
• factor problems for me
• free worksheets combination and permutations
• maths discriminant
• solving a quadratic formula on excel
• 5th Grade math problems for math honors
• hands on activities for graphing system of equalities
• Comic Book Math Worksheets
• c questions aptittude
• how solve approximate integration simpson's rule using matlab
• what do the solutions of a quadratic equation tell you about its graph?
• Simplification of algebraic expression
• Printable worksheets for the 7th grade STAR test
• yr 8 math activity
• finding log using TI 89
• integration by algebric substitution
• math problems for ks3
• N.Y. grade 10 algebra tests
• simplifying polynomials calculator
• simplify rational expressions calculator
• TI 84+ factoring
• find roots of 3rd order polynomial
• printable advanced algebra factoring worksheets
• permutations project
• free printable equations worksheets
• ti-84 plus program formulas pre algebra 1
• numbers poems
• how to solve quadratic function
• greatest common denominator worksheets
• learning algerbra
• slope intercept excel chart
• square root examples
• algebra 2 pics
• algebra tutoring
• past exam papers for ks2
• factoring worksheets using distributive property
• solving homogeneous equations in matlab
• how to simplify +fractions TI-83 Plus Silver Edition
• how to calculate binomial series
• what are the rules for dividing positives and negatives
• easy ways to remember integers
• free pre algebra printouts
• free gcse maths worksheets
• pre algebra slope printouts
• glencoe course 2 math book homework answers
• usable graphing calculator
• solve my rational expression
• materials cost Houghton Mifflin math california edition 5th grade
• get free pre-Algebra worksheets with answer keys worksheets only with averages and rates ratios
• question bank on biology for singapore o level
• what is the number 6 in base 3
• free college algebra pretest
• java+linear+code
• grade 8, algebraic equations, worksheets, isolating variable
• algebra 2 for dummies probability
• second grade pictographs free worksheets
• 3 base number chart
• examples of yr 9 algebra maths tests
• dot math game - printable integer worksheets
• simultaneous equations in excel
• www.larsons math.com
• math test paper print
• logbase ti-89
• online homework help- Summation Notation
• graph an ellipse calculator
• linear equalities
• FL elementary 6th grade math level
• time conversion into decimals
• changing mixed numbers to decimals
• free online tutor practice year 6 sats literacy
• algebraic fractions calculator
• polar coordinate graphing
• how do you calculate the vertex of a parabola
• expressing area of rectangles as a simplified polynomial
• do practice test for the foil method of factoring
• how do i enter function notation on a ti-83 plus graphing calculator
• subtract fractions different denominators worksheet
• square metres coversion to cubic metres
• 9th grade science taks test practice
• substitution and elimination grade 10 math
• Algebra 2 probabilities
• printable homework logs
• FACTORING AND EQUATIONS
• how can i home school algebra two
• multiple variable equations
• nonlinear differential equation
• math word problems.com
• vb6 code to find square root
• programme for TI-84 plus input
• how to solve logs with ti-83
• answer key for algebra for college students fourth edition
• introducing factorization in algebra
• pre-algebra problems free samples
• scale rates + free worksheets
• free algebra worksheets
• free worksheets solving systems of equations
• 6th grade NC EOG math review
• ti 85 ellipse
• "math dilations"
• solution of exercises in book basic algebra by jacobson
• prentice hall pre algebra math activity
• "The Problem SOlver 6" Solutions
• list of math formula
• fraction equations and adding a whole number interactive
• subtracting negative numbers + worksheet
• factoring trinomials
• "A first course in abstract algebra Solutions Manual "
• trinomials calculator
• math answers Geometry McDougal Littell textbooks
• glencoe algebra 2 enrichment answers
• how is Pythagoras work usa today
• basic tutoriels on chemistry
• math ks2
• second order +algibra equation
• convert complex square root numbers to excel formula
• practice 9th grade texas math taks
• maths worksheets on polynomials
• flow chart for algebra mathematical equation
• fastest way to calculate convert decimal to binary kumon
• probability of getting the sum of 6,7 or 8 onthe two dice
• basic algerbra
• solving nonhomogeneous differential equation
• the printable answers of the 8th grade algebra 1 math book problems
• solving one step equations powerpoint
• prentice hall mathematics algebra 2 answers
• cubic function using the TI-84 calculator
• difference of two square rule
• probability of given event in algebra
• phoenix game for calculator
• beginning algebra solver
• glencoe study guide advanced algebra
• grade 3 worksheet term 3
• convert java time
• sats papers ks2-print off
• Holt Algebra I
• 5th Grade Pre Algebra problems
• math homework help polynomial
• sample questions math pi
• java decimal number digits
• Sapmle aptitude questions and answers
• coordinate planes plane worksheet
• graphing a parabola programs on line
• grade 11 polynomial and inequalities
• simple math 11 questions to answer online
• algebra domain calculator
• how to find lowest common multiple for a fraction in vba
• active physics activities printable worksheets
• sats ks2 year 5 past exam papers to do online
• sample aptitude test papers
• free worksheet symmetry in math grade 3
• printable workbook sheets for kids
• dividing and multiplying expressions calc
• adding and subtracting integers worksheet
• simplify cube roots
• algebraTOR
• cube roots worksheets
• math variables work sheet
• algebra step by step solver
• graph ellipse calculator online
• nth roots of fractions
• online 3rd order polynomial solver, free, free
• projects + math + "greatest common factor"
• mixed number algebra equations
• exercise add math-chapter 3 form 5
• Glencoe/McGraw-Hill Algebra worksheet
• permutation combination worksheet
• factorization online
• multiply rational expressions calculator
• sample kumon placement tests
• third root
• runge kutta systems differential equations matlab
• activities with scale factor
• Cramer's Rule practice
• simultaneous equation solver
• a poem about math TAKS exam
• MATH TAKS PREPARATION 3RD GRADERS
• pre algebra worksheets for middle school students
• dividing,adding,multiplying,and subtracting decimals and how they are different
• free t1-85 calculator programs script
• inequality worksheets
• calculator slope and y-intercept
• monomial applets
• ti 84 silver integral solver application
• matlab solve numerically
• matlab combination math
• free sample of math aptitude test
• ks3 maths worksheet
• what does base six mean?
• maths ks3 algebra-games
• matlab solve differential equation
• 9th grade algebra 1 high school math examples
• english aptitude questions
• california 4th grade test prep
• free worksheet on right angles elementary
• pythagoras online calculator
• examples of math trivia with answers
• online math practice for star test
• mathematical glencoe textbook online free
• trigonometry objective online test
• equation of a curve line
• evaluating and simplifying expressions
• pdf form basic math work sheet
• how to solve logs with ti83
• algebra cheats solve equations free
• lineal interpolation calculator
• solved exercises on differential equation
• GCSE quiz
• where to get free teachers edition algebra 1 book answers
• intermediate algebra cheat sheet
• free pre algebra for dummies teaching
• 6th grade algebra equations practice
• summation notation and roots
• java ti84
• math problem solver for mac
• 6th grade cumulative math test
• math solving programs
• linear equations matrix algebrator
• rational expression calculator
• solve second order differential equation nonhomogeneous constant
• Greatest and least common factor worksheets
• basic algerba study guide
• free fun calculator worksheets
• hyperbolic cosine in ti 83
• lowest denominator calculator for 3
• math factors calculator
• nonlinear simultaneous equations
• ti 89 tutorial square root
• how to find square root on a calculator
• ti 84 all equations
• I need FREE, PRINTABLE SOL tests.
• past exams papers for matric mathematics for grade 10
• algebra symbol worksheet
• free online math +work +sheets +for +ged
• solving equations cubed
• math aptitude test
• how to solve homogenous linear equation
• year 8 algebra substitution practise sheet answers
• proportions worksheets +middle school
• square route in excel
• factoring a cubed function
• evaluate square root equations
• free worksheets beginning positive and negative numbers
• Lesson Plans for Teaching Equations to Fifth Graders
• TI 89 log
• the exam in slope problems
• calculator squared, cubed
• online advanced math calculator for rational exponent
• polynomial simplification (x-y)^3
• 1998 sats maths paper
• calculating arcsin on ti 83
• pdf math problems with answers
• A First Course in Abstract Algebra free ebook
• coordinates worksheet
• solving nonlinear difference equations
• symmetry ks2 free samples
• parabola formula
• solving throw formulas
• hands on activities on trigonometry for 4th graders
• finding areas using algebra tiles
• convert mixed numbers and fractions into decimals
• order of operations with TI calculator
• elementary and intermediate algebra by mark dugopolski
• turning a decimal into a fraction calculator
• 10th standard partial fraction formulae
• least common factor of 54 and 35
• linear equation games
• "fraction review worksheets"
• Matlab Powell-Hybrid
• algebra ebook 10th grade free
• factoring calculators
• Quadratics Used in Real Life
• help with elementary algebra 101
• Algebra Tiles Worksheets
• long divition with polynomials calculator
• Ks2 ALGEBRA
• math poem borrowing
• Holt Algebra 1
• algrebra worksheet
• chapter 10 quiz math glencoe answers
• free online trinomial calculator
• free calculators for algebra
• calculate dividends using accounting equation
• fraleigh linear algebra manual solution
• Differences between adding and multiplying
• college math for dummies
• algebra help scientific notation
• integrated algebra workbook 9th grade
• solving quadratics algebraically definition and examples
• algebra calculator expressions
• foil difference of two squares worksheet

Bing users found our website today by using these keyword phrases :

## Example 1: Compound Inequalities

Solve and graph the following inequality: 10 < 2x + 8 < 16

Notice that this is a "conjunction" inequality. Conjunction inequalities usually include the word "and".

If we were to read this aloud, we would say, "10 is less than 2x + 8 AND 2x+8 is less than 16". Notice how the word "and" separates this into two inequalities?

There are actually two different ways to solve this inequality. This first method is to separate this into two different inequalities and solve each independently. The second method is to solve both parts at the same time. I will show you both ways and you can decide which is best for you.

### Method 1: Solve Two Inequalities Independently

For this method, we will use the word "and" to write two different inequalities.

Take note of how the expression in the middle is used twice in order to write two different inequalities.