2.2E: Exercises - Solving Linear Inequalities in Two Variables

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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.

1. Determine whether each ordered pair is a solution to the inequality $y>x−1$:

a. $(0,1)$
b. $(−4,−1)$
c. $(4,2)$
d. $(3,0)$
e. $(−2,−3)$

Answer: a. yes b. yes c. no d. no e. no

2. Determine whether each ordered pair is a solution to the inequality $y \leq 3x+2$:

a. $(0,3)$
b. $(−2,0)$
c. $(-3,-2)$
d. $(0,0)$
e. $(−2,-4)$

Answer: a. no b. yes c. no d. yes e. yes

3. Determine whether each ordered pair is a solution to the inequality $3x−4y>4$:

a. $(5,1)$
b. $(−2,6)$
c. $(3,2)$
d. $(10,−5)$
e. $(0,0)$

Answer: a. yes b. no c. no d. no e. no

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

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

4. Write the inequality shown by the graph with the boundary line $y=3x−4$.

$This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 1), and (2, 2). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.$

Answer: $y\leq 3x−4$

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

$This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 3), (1, 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.$

Answer: $y\geq -x+3$

6. Write the inequality shown by the shaded region in the graph with the boundary line $3x−y=6$.

$This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 6), (1, negative 3), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.$

Answer: $3x−y\leq 6$

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

Answer: $x+y \geq 3$

Graph Linear Inequalities in Two Variables

In the following exercises, graph each linear inequality.

8. Graph the linear inequality: $y>\frac{2}{3}x−1$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 1), (3, 1), and (6, 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.$

9. Graph the linear inequality: $y\geq −\frac{1}{2}x+4$.

Answer: $This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 4), (2, 3), and (4, 2). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.$

10. Graph the linear inequality: $x−y < 3$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (1, negative 2), and (3, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.$

11. Graph the linear inequality: $4x+y>−4$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (negative 1, 0), and (1, negative 8). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.$

12. Graph the linear inequality: $3x+2y > −6$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 3), (3, negative 5), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.$

13. Graph the linear inequality: $y>4x$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (negative 1, negative 4), and (1, 4). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.$

14. Graph the linear inequality: $x−y<4$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, negative 4), (1, negative 3), and (4, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.$

15. Graph the linear inequality: $y> \frac{3}{2}x$.

Answer: $This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 0), (2, 3), and (negative 2, negative 3). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.$

16. Graph the linear inequality: $2x+y< −4$.

Answer: $This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (1, negative 6), and (negative 2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.$

17. Graph the linear inequality: $2x−5y>10$.

Answer: $This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 2), (5, 0), and (negative 5, negative 4). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.$

Solve Applications using Linear Inequalities in Two Variables

18. Harrison works two part time jobs. One at a gas station that pays $11 an hour and the other is IT troubleshooting for $16.50$16.50an hour. Between the two jobs, Harrison wants to earn at least $330 a week. How many hours does Harrison need to work at each job to earn at least $330?

a. Let x be the number of hours he works at the gas station and let y be the number of hours he works troubleshooting. Write an inequality that would model this situation.

b. Graph the inequality.

c. Find three ordered pairs $(x,y)$ that would be solutions to the inequality. Then, explain what that means for Harrison.

Answer

a. $11x+16.5y\geq 330$
b.

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 35. A line is drawn through the points (0, 20), (15, 10), and (30, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.

c. Answers will vary.

19. The doctor tells Laura she needs to exercise enough to burn 500 calories each day. She prefers to either run or bike and burns 15 calories per minute while running and 10 calories a minute while biking.

a. If x is the number of minutes that Laura runs and y is the number minutes she bikes, find the inequality that models the situation.

b. Graph the inequality.

c. List three solutions to the inequality. What options do the solutions provide Laura?

Answer

a. $15x+10y\geq 500$
b.

$This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 60. A line is drawn through the points (0, 50) and (20, 20). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.$

c. Answers will vary.

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