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Contemporary Mathematics

5.10 Systems of Linear Inequalities in Two Variables

Contemporary Mathematics5.10 Systems of Linear Inequalities in Two Variables

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. 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
    13. Chapter 13
  17. Index
An outside view of a coffee shop. People are seen inside sitting and drinking coffee.
Figure 5.87 Many college students find part-time jobs at places such as coffee shops to help pay for college. (credit: modification of work “TULLY’s COFFEE” by MIKI Yoshihito/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Demonstrate whether an ordered pair is a solution to a system of linear inequalities.
  2. Solve systems of linear inequalities using graphical methods.
  3. Graph systems of linear inequalities.
  4. Interpret and solve applications of linear inequalities.

In this section, we will learn how to solve systems of linear inequalities in two variables. In Systems of Linear Equations in Two Variables, we learned how to solve for systems of linear equations in two variables and found a solution that would work in both equations. We can solve systems of inequalities by graphing each inequality (as discussed in Graphing Linear Equations and Inequalities) and putting these on the same coordinate system. The double-shaded part will be our solution to the system. There are many real-life examples for solving systems of linear inequalities.

Consider Ming who has two jobs to help her pay for college. She works at a local coffee shop for $7.50 per hour and at a research lab on campus for $12 per hour. Due to her busy class schedule, she cannot work more than 15 hours per week. If she needs to make at least $150 per week, can she work seven hours at the coffee shop and eight hours in the lab?

Determining If an Ordered Pair Is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is similar to the definition of a system of linear equations. A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.

{ x+4y103x2y<12{ x+4y103x2y<12

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs (x,y)(x,y) that make both inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the xyxy-coordinate system that includes all the points whose ordered pairs make the inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Example 5.89

Determining Whether an Ordered Pair Is a Solution to a System

Determine whether the ordered pair is a solution to the system:

{x+4y103x2y<12{x+4y103x2y<12
  1. (2,4)(2,4)
  2. (3,1)(3,1)

Your Turn 5.89

Determine whether the ordered pair is a solution to the system:
{ x 5 y > 10 2 x + 3 y > 2
1.
( 3 , 1 )
2.
( 6 , 3 )

Solving Systems of Linear Inequalities Using Graphical Methods

The solution to a single linear inequality was the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. We will review graphs of linear inequalities and solve the linear inequality from its graph.

Example 5.90

Solving a System of Linear Inequalities by Graphing

Use Figure 5.88 to solve the system of linear inequalities:

{y2x1y<x+1{y2x1y<x+1
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 6 to 6, in increments of 1. The first (dashed) line passes through the points, (negative 6, negative 5), (negative 1, 0), (0, 1), and (5, 6). The region below the line is shaded in light blue. The second (solid) line passes through the points, (negative 2, negative 5), (0, negative 1), and (4, 7). The region above the line is shaded in dark blue. The two lines intersect approximately at (2, 3). The region below the intersection point and within the lines is shaded in gray.
Figure 5.88

Your Turn 5.90

1.
Use the graph shown to solve the system of linear inequalities:
{ y < 3 x + 2 y > x 1
Two dashed lines are plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The first line passes through the points, (negative 9, 8), (negative 1, 0), (0, 1), and (8, negative 9). The region above the line is shaded in gray. The second line passes through the points, (negative 3, negative 7), (0, 2), and (2, 8). The region to the right of the line is shaded in dark blue. The two lines intersect approximately at (negative 0.8, negative 0.2). The region to the right of the intersection point and within the lines is shaded in light blue.

Graphing Systems of Linear Inequalities

We learned that the solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region by graphing, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Step 1: Graph the first inequality.

Graph the boundary line.

Shade in the side of the boundary line where the inequality is true.

Step 2: On the same grid, graph the second inequality.

Graph the boundary line.

Shade in the side of that boundary line where the inequality is true.

Step 3: The solution is the region where the shading overlaps.

Step 4: Check by choosing a test point.

Example 5.91

Solving a System of Linear Inequalities by Graphing

Solve the system by graphing:

{xy>3y<15x+4{xy>3y<15x+4

Your Turn 5.91

1.
Solve the system by graphing:
{ x + y 2 y 2 3 x 1

Example 5.92

Graphing a System of Linear Inequalities

Solve the system by graphing:

{x2y<5y>4{x2y<5y>4

Your Turn 5.92

1.
Solve the system by graphing:
{ y 3 x 2 y < 1

Systems of linear inequalities where the boundary lines are parallel might have no solution. We will see this in the next example.

Example 5.93

Graphing Parallel Boundary Lines with No Solution

Solve the system by graphing:

{4x+3y12y<43x+1{4x+3y12y<43x+1

Your Turn 5.93

1.
Solve the system by graphing:
{ 3 x 2 y 12 y 3 2 x + 1

Some systems of linear inequalities where the boundary lines are parallel will have a solution. We will see this in the next example.

Example 5.94

Graphing Parallel Boundary Lines with a Solution

Solve the system by graphing:

{y>12x4x2y<4{y>12x4x2y<4

Your Turn 5.94

1.
Solve the system by graphing:
{ y 3 x + 1 3 x + y 4

Interpreting and Solving Applications of Linear Inequalities

When solving applications of systems of inequalities, first translate each condition into an inequality. Then graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so add inequalities to the system as additional requirements.

Example 5.95

Applying Linear Inequalities to Calculating Photo Costs

A photographer sells their prints at a booth at a street fair. At the start of the day, they want to have at least 25 photos to display at their booth. Each small photo they display costs $4 and each large photo costs $10. They do not want to spend more than $200 on photos to display.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could they display 10 small and 20 large photos?
  4. Could they display 20 large and 10 small photos?

Your Turn 5.95

Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn’t want to spend more than $5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs $1.40. Each cookie has 160 calories and costs $0.50.
1.
Write a system of inequalities to model this situation.
2.
Graph the system.
3.
Could he eat 3 hamburgers and 2 cookies?
4.
Could he eat 2 hamburgers and 4 cookies?

Check Your Understanding

Match the correct graph to its system of inequalities.
75.
{ 2 x + 3 y < 5 x y > 3
  1. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, 7.5), (0, 1.5), and (9, negative 4.5). The region below the line is shaded in dark blue. The second line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region above the line is shaded in grey. The two lines intersect at (4, negative 1). The region below the intersection point and within the lines is shaded in light blue.
  2. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first (dashed) line passes through the points, (negative 8, 7), (0, 1.5), (4, negative 1), and (9, negative 4.5). The region above the line is shaded in gray. The second (solid) line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region below the line is shaded in dark blue. The two lines intersect at (4, negative 1). The region above and to the left of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  3. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 8, 7), (0, 1.5), (4, negative ), and (9, negative 4.5). The region above the line is shaded in dark blue. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (9, 6). The region above the line is shaded in light blue. The two lines intersect at (2.8, negative 0.2). The region above the intersection point and within the lines is shaded in gray. Note: all values are approximate.
  4. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 6, 9), (0, 3) (3, 0), and (9, negative 6). The region to the left of the line is shaded in dark blue. The second line passes through the points, (negative 8, negative 7), (0, negative 1.5), (2.5, 0), and (10, 5). The region below the line is shaded in gray. The two lines intersect at (2.8, 0.2). The region below the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  5. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, negative 7), (0, negative 1.5), and (10, 5). The region above the line is shaded in gray. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (10, 7). The region below the line is shaded in dark blue. The two lines intersect at (4, 1). The region to the right of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
76.
{ 2 x 3 y 5 x y 3
  1. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, 7.5), (0, 1.5), and (9, negative 4.5). The region below the line is shaded in dark blue. The second line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region above the line is shaded in grey. The two lines intersect at (4, negative 1). The region below the intersection point and within the lines is shaded in light blue.
  2. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first (dashed) line passes through the points, (negative 8, 7), (0, 1.5), (4, negative 1), and (9, negative 4.5). The region above the line is shaded in gray. The second (solid) line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region below the line is shaded in dark blue. The two lines intersect at (4, negative 1). The region above and to the left of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  3. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 8, 7), (0, 1.5), (4, negative ), and (9, negative 4.5). The region above the line is shaded in dark blue. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (9, 6). The region above the line is shaded in light blue. The two lines intersect at (2.8, negative 0.2). The region above the intersection point and within the lines is shaded in gray. Note: all values are approximate.
  4. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 6, 9), (0, 3) (3, 0), and (9, negative 6). The region to the left of the line is shaded in dark blue. The second line passes through the points, (negative 8, negative 7), (0, negative 1.5), (2.5, 0), and (10, 5). The region below the line is shaded in gray. The two lines intersect at (2.8, 0.2). The region below the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  5. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, negative 7), (0, negative 1.5), and (10, 5). The region above the line is shaded in gray. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (10, 7). The region below the line is shaded in dark blue. The two lines intersect at (4, 1). The region to the right of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
77.
{ 2 x 3 y > 5 x + y < 3
  1. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, 7.5), (0, 1.5), and (9, negative 4.5). The region below the line is shaded in dark blue. The second line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region above the line is shaded in grey. The two lines intersect at (4, negative 1). The region below the intersection point and within the lines is shaded in light blue.
  2. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first (dashed) line passes through the points, (negative 8, 7), (0, 1.5), (4, negative 1), and (9, negative 4.5). The region above the line is shaded in gray. The second (solid) line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region below the line is shaded in dark blue. The two lines intersect at (4, negative 1). The region above and to the left of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  3. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 8, 7), (0, 1.5), (4, negative ), and (9, negative 4.5). The region above the line is shaded in dark blue. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (9, 6). The region above the line is shaded in light blue. The two lines intersect at (2.8, negative 0.2). The region above the intersection point and within the lines is shaded in gray. Note: all values are approximate.
  4. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 6, 9), (0, 3) (3, 0), and (9, negative 6). The region to the left of the line is shaded in dark blue. The second line passes through the points, (negative 8, negative 7), (0, negative 1.5), (2.5, 0), and (10, 5). The region below the line is shaded in gray. The two lines intersect at (2.8, 0.2). The region below the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  5. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, negative 7), (0, negative 1.5), and (10, 5). The region above the line is shaded in gray. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (10, 7). The region below the line is shaded in dark blue. The two lines intersect at (4, 1). The region to the right of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
78.
{ 2 x + 3 y > 5 x + y 3
  1. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, 7.5), (0, 1.5), and (9, negative 4.5). The region below the line is shaded in dark blue. The second line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region above the line is shaded in grey. The two lines intersect at (4, negative 1). The region below the intersection point and within the lines is shaded in light blue.
  2. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first (dashed) line passes through the points, (negative 8, 7), (0, 1.5), (4, negative 1), and (9, negative 4.5). The region above the line is shaded in gray. The second (solid) line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region below the line is shaded in dark blue. The two lines intersect at (4, negative 1). The region above and to the left of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  3. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 8, 7), (0, 1.5), (4, negative ), and (9, negative 4.5). The region above the line is shaded in dark blue. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (9, 6). The region above the line is shaded in light blue. The two lines intersect at (2.8, negative 0.2). The region above the intersection point and within the lines is shaded in gray. Note: all values are approximate.
  4. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 6, 9), (0, 3) (3, 0), and (9, negative 6). The region to the left of the line is shaded in dark blue. The second line passes through the points, (negative 8, negative 7), (0, negative 1.5), (2.5, 0), and (10, 5). The region below the line is shaded in gray. The two lines intersect at (2.8, 0.2). The region below the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  5. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, negative 7), (0, negative 1.5), and (10, 5). The region above the line is shaded in gray. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (10, 7). The region below the line is shaded in dark blue. The two lines intersect at (4, 1). The region to the right of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
79.
{ 2 x + 3 y 5 x + y 3
  1. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, 7.5), (0, 1.5), and (9, negative 4.5). The region below the line is shaded in dark blue. The second line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region above the line is shaded in grey. The two lines intersect at (4, negative 1). The region below the intersection point and within the lines is shaded in light blue.
  2. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first (dashed) line passes through the points, (negative 8, 7), (0, 1.5), (4, negative 1), and (9, negative 4.5). The region above the line is shaded in gray. The second (solid) line passes through the points, (negative 6, 9), (0, 3), (3, 0), and (9, negative 6). The region below the line is shaded in dark blue. The two lines intersect at (4, negative 1). The region above and to the left of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  3. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 8, 7), (0, 1.5), (4, negative ), and (9, negative 4.5). The region above the line is shaded in dark blue. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (9, 6). The region above the line is shaded in light blue. The two lines intersect at (2.8, negative 0.2). The region above the intersection point and within the lines is shaded in gray. Note: all values are approximate.
  4. Two dashed lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 6, 9), (0, 3) (3, 0), and (9, negative 6). The region to the left of the line is shaded in dark blue. The second line passes through the points, (negative 8, negative 7), (0, negative 1.5), (2.5, 0), and (10, 5). The region below the line is shaded in gray. The two lines intersect at (2.8, 0.2). The region below the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
  5. Two lines are plotted on a coordinate plane. The horizontal and vertical axes range from negative 10 to 9, in increments of 1. The first line passes through the points, (negative 9, negative 7), (0, negative 1.5), and (10, 5). The region above the line is shaded in gray. The second line passes through the points, (negative 6, negative 9), (0, negative 3), (3, 0), and (10, 7). The region below the line is shaded in dark blue. The two lines intersect at (4, 1). The region to the right of the intersection point and within the lines is shaded in light blue. Note: all values are approximate.
For the following exercises, determine whether each ordered pair is a solution to the system.
1 .
{ 3 x + y > 5 2 x y 10
A: ( 3 , 3 )
B: ( 7 , 1 )

2 .
{ y < 3 2 x + 3 3 4 x 2 y < 5
A: ( 4 , 1 )
B: ( 8 , 3 )

3 .
{ 4 x y < 10 2 x + 2 y > 8
A: ( 5 , 2 )
B: ( 1 , 3 )

4 .
{ y > 2 3 x 5 x + 1 2 y 4
A: ( 6 , 4 )
B: ( 3 , 0 )

5 .
{ 6 x 5 y < 20 2 x + 7 y > 8
A: ( 1 , 3 )
B: ( 4 , 4 )

6 .
{ 7 x + 2 y > 14 5 x y 8
A: ( 2 , 3 )
B: ( 7 , 1 )

For the following exercises, determine whether each ordered pair is a solution to the darkest shaded region of the graph.
7 .
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first (solid) line passes through the points, (negative 6, 5), (negative 1, 0), (0, negative 1), and (6, negative 7). The region above the line is shaded in blue. The second (dashed) line passes through the points, (negative 2, 6), (0, 2), (2, negative 2), and (4, negative 6). The region below the line is shaded in red. The two lines intersect at (3, negative 4). The region above the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 6 , 8 )
8 .
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first (solid) line passes through the points, (negative 8, 7), (0, 2), (3, 0), and (6, negative 2). The region above the line is shaded in red. The second (dashed) line passes through the points, (negative 2, negative 7.5), (0, negative 3.5), (1.5, 0), and (4, 5). The region to the left of the line is shaded in blue. The two lines intersect at (2, 0.5). The region above the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 2 , 4 )
9 .
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first (solid) line passes through the points, (negative 2, negative 6), (2, 0), and (6, 6). The region to the left of the line is shaded in red. The second (dashed) line passes through the points, (negative 6, 8), (negative 2, 0), (0, negative 4), and (2, negative 8). The region to the left of the line is shaded in blue. The two lines intersect at (negative 3.5, negative 0.5). The region to the left of the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 2 , 2 )
10 .
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first (solid) line passes through the points, (negative 6, 6), (0, 3), and (6, 0). The region below the line is shaded in red. The second (dashed) line is horizontal and it passes through y equals 1. The region below the line is shaded in blue. The two lines intersect at (4, 1). The region below the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 2 , 3 )
11 .
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first line passes through the points, (negative 4.5, 8), (0, 5), and (8, 0). The region below the line is shaded in red. The second line is vertical and it passes through x equals 3. The region to the right of the line is shaded in blue. The two lines intersect at (3, 3). The region below and to the right of the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 6 , 0 )
12 .
Two dashed lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first line passes through the points, (negative 8, 6), (0, 2), (4, 0), and (8, negative 2). The region below the line is shaded in red. The second line passes through the points, (negative 6, negative 8), (0, negative 2), (2, 0), and (8, 6). The region to the right and below the line is shaded in blue. The two lines intersect at (2.5, 0.5). The region below and to the right of the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 6 , 2 )
13 .
Two lines are plotted on an x y coordinate plane. The x and y axes range from negative 8 to 8, in increments of 2. The first (dashed) line passes through the points, (negative 6, 6), (negative 3, 0), and (0, negative 6). The region to the right of the line is shaded in red. The second (solid) line passes through the points, (negative 8, negative 6), (0, negative 2), (4, 0), and (8, 2). The region above the line is shaded in blue. The two lines intersect at (negative 1.5, negative 2.5). The region above and to the right of the intersection point and within the lines is shaded in both colors and it appears dark. Note: all values are approximate.
A: ( 0 , 0 )
B: ( 2 , 2 )
For the following exercises, solve the systems of linear equations by graphing.
14 .
{ y 3 x + 2 y > x 1
15 .
{ y < 2 x 1 y 1 2 x + 4
16 .
{ x y > 1 y < 1 4 x + 3
17 .
{ 3 x y 6 y 1 2 x
18 .
{ 2 x + 4 y 8 y 3 4 x
19 .
{ 2 x 5 y < 10 3 x + 4 y 12
20 .
{ 2 x + 2 y > 4 x + 3 y 9
21 .
{ x 2 y < 3 y 1
22 .
{ x 3 y > 4 y 1
23 .
{ y 1 2 x 3 x 2
24 .
{ x 3 y 6 y > 1 3 x + 1
25 .
{ y 3 4 x 2 y < 2
26 .
{ 3 x 4 y < 8 x < 1
27 .
{ 3 x + 5 y > 10 x > 1
28 .
{ x 3 y 2
29 .
{ x 1 y 3
30 .
{ 2 x + 4 y > 4 y 1 2 x 2
For the following exercises, translate to a system of inequalities and solve.
A gardener does not want to spend more than $50 on bags of fertilizer and peat moss for their garden. Fertilizer costs $2 a bag and peat moss costs $5 a bag. The gardener’s van can hold at most 20 bags.
31 .
Write a system of inequalities to model this situation.
32 .
Graph the system.
33 .
Can they buy 15 bags of fertilizer and 4 bags of peat moss?
34 .
Can they buy 10 bags of fertilizer and 10 bags of peat moss?
For the following exercises, translate to a system of inequalities and solve.
A student is studying for their final exams in chemistry and algebra. They only have 24 hours to study, and it will take them at least 3 times as long to study for algebra than chemistry.
35 .
Write a system of inequalities to model this situation.
36 .
Graph the system.
37 .
Can they spend 4 hours on chemistry and 20 hours on algebra?
38 .
Can they spend 6 hours on chemistry and 18 hours on algebra?
For the following exercises, translate to a system of inequalities and solve.
Mara is attempting to build muscle mass. To do this, she needs to eat an additional 80 grams of protein or more in a day. A bottle of protein water costs $3.20 and a protein bar costs $1.75. The protein water supplies 27 grams of protein and the bar supplies 16 grams. Let w be the number of water bottles Mara can buy, and let b be the number of protein bars she can buy. If Mara has $10 dollars to spend:
39 .
Write a system of inequalities to model this situation.
40 .
Graph the system.
41 .
Could she buy 3 bottles of protein water and 1 protein bar?
42 .
Could she buy no bottles of protein water and 5 protein bars?
For the following exercises, translate to a system of inequalities and solve.
Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1,500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories/mile.
43 .
Write a system of inequalities to model this situation.
44 .
Graph the system.
45 .
Could he meet his goal by walking 3 miles and running 1 mile?
46 .
Could he meet his goal by walking 2 miles and running 2 miles?
For the following exercises, translate to a system of inequalities and solve.
Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend on the extra food he needs and will spend it on $0.75 donuts, which have 360 calories each, and $2 energy drinks, which have 110 calories.
47 .
Write a system of inequalities that models this situation.
48 .
Graph the system.
49 .
Can he buy 8 donuts and 4 energy drinks and satisfy his caloric needs?
50 .
Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs?
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