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Solving Linear Inequalities in Mathematics: Examples and Exercises - Prof. John Fetcho, Exams of Algebra

A part of math 123 section 2.7, focusing on solving linear inequalities in one variable. It includes examples and exercises to help students understand the concept. Students are required to define a variable, write an equation/inequality, solve it, and answer the question with a complete sentence.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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Math 123 – Section 2.7 – Solving Linear Inequalities – Part II
Page 1
© Copyright 2009 by John Fetcho. All rights reserved
Section 2.7
Linear Inequalities in One Variable – Part II
I. When solving word problems, remember to:
A. Define a variable for the unknown.
B. Write an equation/inequality expressing the situation.
C. Solve.
D. Answer the question with a complete sentence.
E. Examples - Solve each of the following.
1. The bar graph (top of right-hand column, page 184) shows the percentage of
wages full-time workers pay in income tax in eight selected countries. (The
percents shown are averages for single earners without children.) Let x
represent the percentage of wages workers pay in income tax. In Exercises
99-104, write the name of the country or countries described by the given
inequality.
x > 30% (#100)
We want to find all the countries greater than 30%, not included.
Answer: {Netherlands, Denmark}
2. Now you try one: 25% < x < 40% (#103)
Answer: {United States, Canada, Norway, Netherlands}
3. The line graph (middle of right-hand column, page 184) shows the declining
consumption of cigarettes in the United States. The data shown by the graph
can be modeled by
N = 550 – 9x
where N is the number of cigarettes consumed, in billions, x years after 1988.
Use this formula to solve Exercises 105 – 106.
Describe how many years after 1988 cigarette consumption will be less than
325 billion cigarettes per year. Which years are included in your description?
(#106)
Let N = 325 Remember that N is already measured in billions.
Let x = The number of years after 1988.
So x = 0 means 1988
x = 1 means 1989
x = 2 means 1990
x = 3 means 1991
etc.
So x = 17 means 2005
We now need to substitute in for N and write an inequality.
550 – 9x < 325 Subtract 550 from each side to isolate the variable term.
9x < 225 Divide both sides by 9 to isolate the variable. Remember
to switch the inequality sign!
x > 25 Add to 1988 to determine the year.
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Math 123 – Section 2.7 – Solving Linear Inequalities – Part II − Page 1

© Copyright 2009 by John Fetcho. All rights reserved

Section 2.

Linear Inequalities in One Variable – Part II

I. When solving word problems, remember to: A. Define a variable for the unknown. B. Write an equation/inequality expressing the situation. C. Solve. D. Answer the question with a complete sentence.

E. Examples - Solve each of the following.

  1. The bar graph (top of right-hand column, page 184) shows the percentage of wages full-time workers pay in income tax in eight selected countries. (The percents shown are averages for single earners without children.) Let x represent the percentage of wages workers pay in income tax. In Exercises 99-104, write the name of the country or countries described by the given inequality.

x > 30% (#100)

We want to find all the countries greater than 30%, not included.

Answer: {Netherlands, Denmark}

  1. Now you try one: 25% < x < 40% (#103)

Answer: {United States, Canada, Norway, Netherlands}

  1. The line graph (middle of right-hand column, page 184) shows the declining consumption of cigarettes in the United States. The data shown by the graph can be modeled by

N = 550 – 9x

where N is the number of cigarettes consumed, in billions, x years after 1988. Use this formula to solve Exercises 105 – 106.

Describe how many years after 1988 cigarette consumption will be less than 325 billion cigarettes per year. Which years are included in your description? (#106)

Let N = 325 Remember that N is already measured in billions. Let x = The number of years after 1988. So x = 0 means 1988 x = 1 means 1989 x = 2 means 1990 x = 3 means 1991 etc. So x = 17 means 2005

We now need to substitute in for N and write an inequality.

550 – 9x < 325 Subtract 550 from each side to isolate the variable term. −9x < − 225 Divide both sides by −9 to isolate the variable. Remember to switch the inequality sign! x > 25 Add to 1988 to determine the year.

Math 123 – Section 2.7 – Solving Linear Inequalities – Part II − Page 2

© Copyright 2009 by John Fetcho. All rights reserved

1988 + 25 = 2013 Answer the question.

Answer: From 2013 on, cigarette consumption will be less that 325 billion cigarettes per year, in the United States. We won’t talk about the rest of the world!

  1. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each bag of cement weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip? (Page 185, #112)

On this problem, we need to have that the sum of the operator’s weight and the weight of the number of bags of cement has be less than or equal to 2800 pounds. So what don’t we know here? The number of bags of cement, of course! So this tells us what to let our variable represent.

Let x = The number of bags of cement. 65x = The weight of the bags of cement.

265 + 65x < 2800 Subtract 265 from each side to isolate the variable term. 65x < 2535 Divide both sides by 65 to isolate the variable. x < 39 Answer the question.

Answer: The elevator can safely lift 39 bags of cement at one time.

  1. Now you try one: On two examinations, you have grades of 86 and 88. There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90. What must you get on the final to earn an A in the course? (Page 184, #107a)

Answer: Inequality: 90 3

86 88 x ≥≥≥≥

; You have to get a 96 on the final to

get an A in the class.