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Intermediate Algebra 2e

3.4 Graph Linear Inequalities in Two Variables

Intermediate Algebra 2e3.4 Graph Linear Inequalities in Two Variables
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Verify solutions to an inequality in two variables.
  • Recognize the relation between the solutions of an inequality and its graph.
  • Graph linear inequalities in two variables
  • Solve applications using linear inequalities in two variables
Be Prepared 3.10

Before you get started, take this readiness quiz.

Graph x>2x>2 on a number line.
If you missed this problem, review Example 2.48.

Be Prepared 3.11

Solve: 4x+3>234x+3>23.
If you missed this problem, review Example 2.52.

Be Prepared 3.12

Translate: 8<x>38<x>3.
If you missed this problem, review Example 2.56.

Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.

Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

Linear Inequality

A linear inequality is an inequality that can be written in one of the following forms:

Ax+By>CAx+ByCAx+By<CAx+ByCAx+By>CAx+ByCAx+By<CAx+ByC

Where A and B are not both zero.

Recall that an inequality with one variable had many solutions. For example, the solution to the inequality x>3x>3 is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See Figure 3.10.


Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.
Figure 3.10

Similarly, linear inequalities in two variables have many solutions. Any ordered pair (x,y)(x,y) that makes an inequality true when we substitute in the values is a solution to a linear inequality.

Solution to a Linear Inequality

An ordered pair (x,y)(x,y) is a solution to a linear inequality if the inequality is true when we substitute the values of x and y.

Example 3.34

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

(0,0)(0,0) (1,6)(1,6) (2,6)(2,6) (−5,−15)(−5,−15) (−8,12)(−8,12)

Try It 3.67

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

(0,0)(0,0) (4,9)(4,9) (−2,1)(−2,1) (−5,−3)(−5,−3) (5,1)(5,1)

Try It 3.68

Determine whether each ordered pair is a solution to the inequality y<x+1:y<x+1:

(0,0)(0,0) (8,6)(8,6) (−2,−1)(−2,−1) (3,4)(3,4) (−1,−4)(−1,−4)

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

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in shown previously again. The point x=3x=3 separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See Figure 3.11.

Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled “numbers greater than 3”. The part of the number line to the left of 3 is labeled “numbers less than 3”.
Figure 3.11 The solution to x>3x>3 is the shaded part of the number line to the right of x=3.x=3.

Similarly, the line y=x+4y=x+4 separates the plane into two regions. On one side of the line are points with y<x+4.y<x+4. On the other side of the line are the points with y>x+4.y>x+4. We call the line y=x+4y=x+4 a boundary line.

Boundary Line

The line with equation Ax+By=CAx+By=C is the boundary line that separates the region where Ax+By>CAx+By>C from the region where Ax+By<C.Ax+By<C.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:

Two number lines are shown with the middle labeled with the number “a”. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled “x is less than a” and the number a is marked with an open parenthesis. The second number line is labeled “x is less than or equal to a” and the number a is marked with an open bracket.

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.

Ax+By<CAx+ByC Ax+By>CAx+ByC Boundary line isAx+By=CBoundary line isAx+By=C Boundary line is not included in solution.Boundary line is included in solution. Boundary line is dashed.Boundary line is solid.Ax+By<CAx+ByC Ax+By>CAx+ByC Boundary line isAx+By=CBoundary line isAx+By=C Boundary line is not included in solution.Boundary line is included in solution. Boundary line is dashed.Boundary line is solid.

Now, let’s take a look at what we found in Example 3.34. We’ll start by graphing the line y=x+4,y=x+4, and then we’ll plot the five points we tested, as shown in the graph. See Figure 3.12.

This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6).
Figure 3.12

In Example 3.34 we found that some of the points were solutions to the inequality y>x+4y>x+4 and some were not.

Which of the points we plotted are solutions to the inequality y>x+4?y>x+4?

The points (1,6)(1,6) and (−8,12)(−8,12) are solutions to the inequality y>x+4.y>x+4. Notice that they are both on the same side of the boundary line y=x+4.y=x+4.

The two points (0,0)(0,0) and (−5,−15)(−5,−15) are on the other side of the boundary line y=x+4,y=x+4, and they are not solutions to the inequality y>x+4.y>x+4. For those two points, y<x+4.y<x+4.

What about the point (2,6)?(2,6)? Because 6=2+4,6=2+4, the point is a solution to the equation y=x+4,y=x+4, but not a solution to the inequality y>x+4.y>x+4. So the point (2,6)(2,6) is on the boundary line.

Let’s take another point above the boundary line and test whether or not it is a solution to the inequality y>x+4.y>x+4. The point (0,10)(0,10) clearly looks to above the boundary line, doesn’t it? Is it a solution to the inequality?

y>x+410>?0+410>4y>x+410>?0+410>4

So, (0,10)(0,10) is a solution to y>x+4.y>x+4.

Any point you choose above the boundary line is a solution to the inequality y>x+4.y>x+4. All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with (0,0)(0,0) and (−5,−15),(−5,−15), are not solutions to y>x+4,y>x+4, as shown in Figure 3.13.

This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.
Figure 3.13

The graph of the inequality y>x+4y>x+4 is shown in below.

The line y=x+4y=x+4 divides the plane into two regions. The shaded side shows the solutions to the inequality y>x+4.y>x+4.

The points on the boundary line, those where y=x+4,y=x+4, are not solutions to the inequality y>x+4,y>x+4, so the line itself is not part of the solution. We show that by making the line dashed, not solid.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.

Example 3.35

The boundary line shown in this graph is y=2x1.y=2x1. Write the inequality shown by the graph.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.
Try It 3.69

Write the inequality shown by the graph with the boundary line y=−2x+3.y=−2x+3.

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, 3), (1, 1), and (3, negative 3). The line divides the x y-coordinate plane into two halves. The line itself and the top right half are colored red to indicate that this is where the solutions of the inequality are.
Try It 3.70

Write the inequality shown by the graph with the boundary line y=12x4.y=12x4.

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 4), (2, negative 3), and (4, negative 2). The line divides the x y-coordinate plane into two halves. The line itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.

Example 3.36

The boundary line shown in this graph is 2x+3y=6.2x+3y=6. Write the inequality shown by the graph.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.
Try It 3.71

Write the inequality shown by the shaded region in the graph with the boundary line x4y=8.x4y=8.

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight line is drawn through the points (0, negative 2), (4, negative 1), and (8, 0). The line divides the x y-coordinate plane into two halves. The line itself and the top left half are colored red to indicate that this is where the solutions of the inequality are.
Try It 3.72

Write the inequality shown by the shaded region in the graph with the boundary line 3xy=6.3xy=6.

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight 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 itself and the bottom right half are colored red to indicate that this is where the solutions of the inequality are.

Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

Example 3.37 How to Graph a Linear Equation in Two Variables

Graph the linear inequality y34x2.y34x2.

Try It 3.73

Graph the linear inequality y52x4.y52x4.

Try It 3.74

Graph the linear inequality y23x5.y23x5.

The steps we take to graph a linear inequality are summarized here.

How To

Graph a linear inequality in two variables.

  1. Step 1. Identify and graph the boundary line.
    • If the inequality is or,or, the boundary line is solid.
    • If the inequality is <or>,<or>, the boundary line is dashed.
  2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. Step 3. Shade in one side of the boundary line.
    • If the test point is a solution, shade in the side that includes the point.
    • If the test point is not a solution, shade in the opposite side.

Example 3.38

Graph the linear inequality x2y<5.x2y<5.

Try It 3.75

Graph the linear inequality: 2x3y<6.2x3y<6.

Try It 3.76

Graph the linear inequality: 2xy>3.2xy>3.

What if the boundary line goes through the origin? Then, we won’t be able to use (0,0)(0,0) as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Example 3.39

Graph the linear inequality: y4x.y4x.

Try It 3.77

Graph the linear inequality: y>3x.y>3x.

Try It 3.78

Graph the linear inequality: y−2x.y−2x.

Some linear inequalities have only one variable. They may have an x but no y, or a y but no x. In these cases, the boundary line will be either a vertical or a horizontal line.

Recall that:

x=avertical line y=bhorizontal line x=avertical line y=bhorizontal line

Example 3.40

Graph the linear inequality: y>3.y>3.

Try It 3.79

Graph the linear inequality: y<5.y<5.

Try It 3.80

Graph the linear inequality: y−1.y−1.

Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how thay might be used.

Example 3.41

Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least $240 a week. Her job in food service pays $10 an hour and her tutoring job on campus pays $15 an hour. How many hours does Hilaria need to work at each job to earn at least $240?

Let xx be the number of hours she works at the job in food service and let y be the number of hours she works tutoring. Write an inequality that would model this situation.

Graph the inequality.

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

Try It 3.81

Hugh works two part time jobs. One at a grocery store that pays $10 an hour and the other is babysitting for $13 hour. Between the two jobs, Hugh wants to earn at least $260 a week. How many hours does Hugh need to work at each job to earn at least $260?

Let x be the number of hours he works at the grocery store and let y be the number of hours he works babysitting. Write an inequality that would model this situation.

Graph the inequality.

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

Try It 3.82

Veronica works two part time jobs in order to earn enough money to meet her obligations of at least $280 a week. Her job at the day spa pays $10 an hour and her administrative assistant job on campus pays $17.50 an hour. How many hours does Veronica need to work at each job to earn at least $280?

Let x be the number of hours she works at the day spa and let y be the number of hours she works as administrative assistant. Write an inequality that would model this situation.

Graph the inequality.

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

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with graphing linear inequalities in two variables.

Section 3.4 Exercises

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.

237.

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


(0,1)(0,1)
(−4,−1)(−4,−1)
(4,2)(4,2)
(3,0)(3,0)
(−2,−3)(−2,−3)

238.

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


(0,0)(0,0)
(2,1)(2,1)
(−1,−5)(−1,−5)
(−6,−3)(−6,−3)
(1,0)(1,0)

239.

Determine whether each ordered pair is a solution to the inequality y<3x+2:y<3x+2:


(0,3)(0,3)
(−3,−2)(−3,−2)
(−2,0)(−2,0)
(0,0)(0,0)
(−1,4)(−1,4)

240.

Determine whether each ordered pair is a solution to the inequality y<2x+5:y<2x+5:


(−3,0)(−3,0)
(1,6)(1,6)
(−6,−2)(−6,−2)
(0,1)(0,1)
(5,−4)(5,−4)

241.

Determine whether each ordered pair is a solution to the inequality 3x4y>4:3x4y>4:


(5,1)(5,1)
(−2,6)(−2,6)
(3,2)(3,2)
(10,−5)(10,−5)
(0,0)(0,0)

242.

Determine whether each ordered pair is a solution to the inequality 2x+3y>2:2x+3y>2:


(1,1)(1,1)
(4,−3)(4,−3)
(0,0)(0,0)
(−8,12)(−8,12)
(3,0)(3,0)

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

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

243.

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

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.
244.

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

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 2), and (2, 0). 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.
245.

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

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, 1), (2, 0), and (4, negative 1). 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.
246.

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

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), (3, negative 3), and (6, negative 4). 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.
247.

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

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, 5), (1, 4), and (5, 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.
248.

Write the inequality shown by the shaded region in the graph with the boundary line x+y=3.x+y=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.
249.

Write the inequality shown by the shaded region in the graph with the boundary line 3xy=6.3xy=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.
250.

Write the inequality shown by the shaded region in the graph with the boundary line 2xy=4.2xy=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 2), 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.

Graph Linear Inequalities in Two Variables

In the following exercises, graph each linear inequality.

251.

Graph the linear inequality: y>23x1.y>23x1.

252.

Graph the linear inequality: y<35x+2.y<35x+2.

253.

Graph the linear inequality: y12x+4.y12x+4.

254.

Graph the linear inequality: y13x2.y13x2.

255.

Graph the linear inequality: xy3.xy3.

256.

Graph the linear inequality: xy−2.xy−2.

257.

Graph the linear inequality: 4x+y>4.4x+y>4.

258.

Graph the linear inequality: x+5y<5.x+5y<5.

259.

Graph the linear inequality: 3x+2y−6.3x+2y−6.

260.

Graph the linear inequality: 4x+2y−8.4x+2y−8.

261.

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

262.

Graph the linear inequality: y−3x.y−3x.

263.

Graph the linear inequality: y<10.y<10.

264.

Graph the linear inequality: y2.y2.

265.

Graph the linear inequality: x5.x5.

266.

Graph the linear inequality: x0.x0.

267.

Graph the linear inequality: xy<4.xy<4.

268.

Graph the linear inequality: xy<3.xy<3.

269.

Graph the linear inequality: y32x.y32x.

270.

Graph the linear inequality: y54x.y54x.

271.

Graph the linear inequality: y>2x+1.y>2x+1.

272.

Graph the linear inequality: y<3x4.y<3x4.

273.

Graph the linear inequality: 2x+y−4.2x+y−4.

274.

Graph the linear inequality: x+2y−2.x+2y−2.

275.

Graph the linear inequality: 2x5y>10.2x5y>10.

276.

Graph the linear inequality: 4x3y>12.4x3y>12.

Solve Applications using Linear Inequalities in Two Variables

277.

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$16.50 an 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?

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.

Graph the inequality.

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

278.

Elena needs to earn at least $450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays $9 an hour and the other as an intern in a genetics lab for $22.50 per hour. How many hours does Elena need to work at each job to earn at least $450 per week?

Let x be the number of hours she works teaching swimming and let y be the number of hours she works as an intern. Write an inequality that would model this situation.

Graph the inequality.

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

279.

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.

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.

Graph the inequality.

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

280.

Armando’s workouts consist of kickboxing and swimming. While kickboxing, he burns 10 calories per minute and he burns 7 calories a minute while swimming. He wants to burn 600 calories each day.

If x is the number of minutes that Armando will kickbox and y is the number minutes he will swim, find the inequality that will help Armando create a workout for today.

Graph the inequality.

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

Writing Exercises

281.

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.

282.

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

Self Check

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

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “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 other columns are left blank so that the learner may indicate their mastery level for each topic.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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