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Mathematics LibreTexts

2.10E: Exercises

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Practice Makes Perfect

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Exercise 2.10.1

Determine whether each ordered pair is a solution to the inequality y>x1:

  1. (0,1)
  2. (4,1)
  3. (4,2)
  4. (3,0)
  5. (2,3)
Exercise 2.10.2

Determine whether each ordered pair is a solution to the inequality y>x3:

  1. (0,0)
  2. (2,1)
  3. (1,5)
  4. (6,3)
  5. (1,0)
Answer
  1. yes 
  2. no 
  3. no 
  4. yes 
  5. no
Exercise 2.10.3

Determine whether each ordered pair is a solution to the inequality y<x+2:

  1. (0,3)
  2. (3,2)
  3. (2,0)
  4. (0,0)
  5. (1,4)
Exercise 2.10.4

Determine whether each ordered pair is a solution to the inequality y<x+5:

  1. (3,0)
  2. (1,6)
  3. (6,2)
  4. (0,1)
  5. (5,4)
Answer
  1. yes 
  2. no
  3. no 
  4. yes 
  5. yes
Exercise 2.10.5

Determine whether each ordered pair is a solution to the inequality x+y>4:

  1. (5,1)
  2. (2,6)
  3. (3,2)
  4. (10,5)
  5. (0,0)
Exercise 2.10.6

Determine whether each ordered pair is a solution to the inequality x+y>2:

  1. (1,1)
  2. (4,3)
  3. (0,0)
  4. (8,12)
  5. (3,0)
Answer
  1. no 
  2. no 
  3. no 
  4. yes 
  5. yes

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Exercise 2.10.7

Write the inequality shown by the graph with the boundary line y=3x4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.

Exercise 2.10.8

Write the inequality shown by the graph with the boundary line y=2x4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Answer

y<2x4

Exercise 2.10.9

Write the inequality shown by the graph with the boundary line y=12x+1

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Exercise 2.10.10

Write the inequality shown by the graph with the boundary line y=13x2

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Answer

y13x2

Exercise 2.10.11

Write the inequality shown by the shaded region in the graph with the boundary line x+y=5.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 5 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

Exercise 2.10.12

Write the inequality shown by the shaded region in the graph with the boundary line x+y=3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

Answer

x+y3

Exercise 2.10.13

Write the inequality shown by the shaded region in the graph with the boundary line 2x+y=4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Exercise 2.10.14

Write the inequality shown by the shaded region in the graph with the boundary line x+2y=2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 2 y equals negative 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Answer

x+2y2

Exercise 2.10.15

Write the inequality shown by the shaded region in the graph with the boundary line 3xy=6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

Exercise 2.10.16

Write the inequality shown by the shaded region in the graph with the boundary line 2xy=4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

Answer

2xy<4

Exercise 2.10.17

Write the inequality shown by the shaded region in the graph with the boundary line 2x5y=10.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 5 y equals 10 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

Exercise 2.10.18

Write the inequality shown by the shaded region in the graph with the boundary line 4x3y=12.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

Answer

4x3y>12

Graph Linear Inequalities

In the following exercises, graph each linear inequality.

Exercise 2.10.19

Graph the linear inequality y>23x1

Exercise 2.10.20

Graph the linear inequality y<35x+2

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals three-fifths x plus 2 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

Exercise 2.10.21

Graph the linear inequality y12x+4

Exercise 2.10.22

Graph the linear inequality y13x2

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Exercise 2.10.23

Graph the linear inequality xy3

Exercise 2.10.24

Graph the linear inequality xy2

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 2 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Exercise 2.10.25

Graph the linear inequality 4x+y>4

Exercise 2.10.26

Graph the linear inequality x+5y<5

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 5 y equals negative 5 is plotted as a dashed line extending from the top left toward the bottom right. The region below the line is shaded.

Exercise 2.10.27

Graph the linear inequality 3x+2y6

Exercise 2.10.28

Graph the linear inequality 4x+2y8

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x plus 2 y equals negative 8 is plotted as a solid line extending from the top left toward the bottom right. The region to the right of the line is shaded.

Exercise 2.10.29

Graph the linear inequality y>4x

Exercise 2.10.30

Graph the linear inequality y>x

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x is plotted as a solid line extending from the bottom left toward the top right. The region above the line is shaded.

Exercise 2.10.31

Graph the linear inequality yx

Exercise 2.10.32

Graph the linear inequality y3x

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a solid line extending from the top left toward the bottom right. The region to the left of the line is shaded.

Exercise 2.10.33

Graph the linear inequality y2

Exercise 2.10.34

Graph the linear inequality y<1

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed horizontal line. The region below the line is shaded.

Exercise 2.10.35

Graph the linear inequality y<4

Exercise 2.10.36

Graph the linear inequality y2

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 is plotted as a solid horizontal line. The region above the line is shaded.

Exercise 2.10.37

Graph the linear inequality x5

Exercise 2.10.38

Graph the linear inequality x>2

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 2 is plotted as a dashed vertical line. The region to the right of the line is shaded.

Exercise 2.10.39

Graph the linear inequality x>3

Exercise 2.10.40

Graph the linear inequality x4

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals 4 is plotted as a solid vertical line. The region to the left of the line is shaded.

Exercise 2.10.41

Graph the linear inequality xy<4

Exercise 2.10.42

Graph the linear inequality xy<3

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.

Exercise 2.10.43

Graph the linear inequality y32x

Exercise 2.10.44

Graph the linear inequality y54x

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-fourths x is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Exercise 2.10.45

Graph the linear inequality y>2x+1

Exercise 2.10.46

Graph the linear inequality y<3x4

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x minus 4 is plotted as a dashed line extending from the top left toward the bottom right. The region to the left of the line is shaded.

Exercise 2.10.47

Graph the linear inequality x1

Exercise 2.10.48

Graph the linear inequality x0

Answer

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 0 is plotted as a solid vertical line along the y-axis. The region to the right of the line is shaded.

Everyday Math

Exercise 2.10.49

Money. Gerry wants to have a maximum of $100 cash at the ticket booth when his church carnival opens. He will have $1 bills and $5 bills. If x is the number of $1 bills and y is the number of $5 bills, the inequality x+5y100 models the situation.

  1. Graph the inequality.
  2. List three solutions to the inequality x+5y100 where both x and y are integers.
Exercise 2.10.50

Shopping. Tula has $20 to spend at the used book sale. Hardcover books cost $2 each and paperback books cost $0.50 each. If x is the number of hardcover books Tula can buy and y is the number of paperback books she can buy, the inequality 2x+12y20 models the situation.

  1. Graph the inequality.
  2. List three solutions to the inequality 2x+12y20 where both x and y are whole numbers.
Answer

1.
The graph shows the x y-coordinate plane. The x- axis runs from 0 to 20 and the y-axis runs from 0 to 30. The line 2 x plus one-half y equals 20 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

2. Answers will vary.

Writing Exercises

Exercise 2.10.51

Lester thinks that the solution of any inequality with a > sign is the region above the line and the solution of any inequality with a < sign is the region below the line. Is Lester correct? Explain why or why not.

Exercise 2.10.52

Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “verify solutions to an inequality in two variables,”, “recognize the relation between the solutions of an inequality and its graph,” and “graph linear inequalities.” The rest of the cells are blank.

b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?


2.10E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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