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Elementary Algebra 2e

5.6 Graphing Systems of Linear Inequalities

Elementary Algebra 2e5.6 Graphing Systems of Linear Inequalities
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index
Be Prepared 5.16

Before you get started, take this readiness quiz.

Graph x>2x>2 on a number line.
If you missed this problem, review Example 2.66.

Be Prepared 5.17

Solve the inequality 2a<5a+122a<5a+12.
If you missed this problem, review Example 2.73.

Be Prepared 5.18

Determine whether the ordered pair (3,12)(3,12) is a solution to the system {x+2y=4y=6x{x+2y=4y=6x.
If you missed this problem, review Example 5.1

 

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

System of Linear Inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

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 below.

{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.

Solutions of a System of Linear Inequalities

Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.

The solution of a system of linear inequalities is shown as a shaded region in the x-y 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, we 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 5.51

Determine whether the ordered pair is a solution to the system. {x+4y103x2y<12{x+4y103x2y<12

(−2, 4) (3,1)

Try It 5.101

Determine whether the ordered pair is a solution to the system.
{x5y>102x+3y>−2{x5y>102x+3y>−2

(3,−1)(3,−1) (6,−3)(6,−3)

Try It 5.102

Determine whether the ordered pair is a solution to the system.
{y>4x24xy<20{y>4x24xy<20

(2,1)(2,1) (4,−1)(4,−1)

Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is 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. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Example 5.52

How to Solve a System of Linear inequalities

Solve the system by graphing.

{y2x1y<x+1{y2x1y<x+1

Try It 5.103

Solve the system by graphing. {y<3x+2y>x1{y<3x+2y>x1

Try It 5.104

Solve the system by graphing. {y<12x+3y<3x4{y<12x+3y<3x4

How To

Solve a system of linear inequalities by graphing.

  1. Step 1. Graph the first inequality.
    • Graph the boundary line.
    • Shade in the side of the boundary line where the inequality is true.
  2. 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.
  3. Step 3. The solution is the region where the shading overlaps.
  4. Step 4. Check by choosing a test point.

Example 5.53

Solve the system by graphing. {xy>3y<15x+4{xy>3y<15x+4

Try It 5.105

Solve the system by graphing. {x+y2y23x1{x+y2y23x1

Try It 5.106

Solve the system by graphing. {3x2y6y>14x+5{3x2y6y>14x+5

Example 5.54

Solve the system by graphing. {x2y<5y>−4{x2y<5y>−4

Try It 5.107

Solve the system by graphing. {y3x2y<−1{y3x2y<−1

Try It 5.108

Solve the system by graphing. {x>−4x2y4{x>−4x2y4

Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in Example 5.55.

Example 5.55

Solve the system by graphing. {4x+3y12y<43x+1{4x+3y12y<43x+1

Try It 5.109

Solve the system by graphing. {3x2y12y32x+1{3x2y12y32x+1

Try It 5.110

Solve the system by graphing. {x+3y>8y<13x2{x+3y>8y<13x2

Example 5.56

Solve the system by graphing. {y>12x4x2y<−4{y>12x4x2y<−4

Try It 5.111

Solve the system by graphing. {y3x+1−3x+y−4{y3x+1−3x+y−4

Try It 5.112

Solve the system by graphing. {y14x+2x+4y4{y14x+2x+4y4

Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we 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 their graphs will only show Quadrant I.

Example 5.57

Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her $4 and each large photo costs her $10. She doesn’t want to spend more than $200 on photos to display.

Write a system of inequalities to model this situation.

Graph the system.

Could she display 15 small and 5 large photos?

Could she display 3 large and 22 small photos?

Try It 5.113

A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could 4 microwaves and 2 printers be carried on this trailer?
  4. Could 7 microwaves and 3 printers be carried on this trailer?
Try It 5.114

Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost $2 and the answer sheets cost $1. Mary’s budget for these supplies allows for a maximum cost of $400.

Write a system of inequalities to model this situation.
Graph the system.
Could Mary purchase 100 pencils and 100 answer sheets?
Could Mary purchase 150 pencils and 150 answer sheets?

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could Mary purchase 100 pencils and 100 answer sheets?
  4. Could Mary purchase 150 pencils and 150 answer sheets?

Example 5.58

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 1 cookie?
  4. Could he eat 2 hamburgers and 4 cookies?
Try It 5.115

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.

  1. Write a system of inequalities that models this situation.
  2. Graph the system.
  3. Can he buy 8 donuts and 4 energy drinks?
  4. Can he buy 1 donut and 3 energy drinks?
Try It 5.116

Philip’s doctor tells him he should add at least 1000 more calories per day to his usual diet. Philip wants to buy protein bars that cost $1.80 each and have 140 calories and juice that costs $1.25 per bottle and have 125 calories. He doesn’t want to spend more than $12.

  1. Write a system of inequalities that models this situation.
  2. Graph the system.
  3. Can he buy 3 protein bars and 5 bottles of juice?
  4. Can he buy 5 protein bars and 3 bottles of juice?

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with graphing systems of linear inequalities.

Section 5.6 Exercises

Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

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

275.

{3x+y>52xy10{3x+y>52xy10

(3,−3)(3,−3) (7,1)(7,1)

276.

{4xy<10−2x+2y>−8{4xy<10−2x+2y>−8

(5,−2)(5,−2) (−1,3)(−1,3)

277.

{y>23x5x+12y4{y>23x5x+12y4

(6,−4)(6,−4) (3,0)(3,0)

278.

{y<32x+334x2y<5{y<32x+334x2y<5

(−4,−1)(−4,−1) (8,3)(8,3)

279.

{7x+2y>145xy8{7x+2y>145xy8

(2,3)(2,3) (7,−1)(7,−1)

280.

{6x5y<20−2x+7y>−8{6x5y<20−2x+7y>−8

(1,−3)(1,−3) (−4,4)(−4,4)

281.

{2x+3y24x6y<−1{2x+3y24x6y<−1

(32,43)(32,43) (14,76)(14,76)

282.

{5x3y<−210x+6y>4{5x3y<−210x+6y>4

(15,23)(15,23) (310,76)(310,76)

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

283.

{y3x+2y>x1{y3x+2y>x1

284.

{y<−2x+2yx1{y<−2x+2yx1

285.

{y<2x1y12x+4{y<2x1y12x+4

286.

{y23x+2y>2x3{y23x+2y>2x3

287.

{xy>1y<14x+3{xy>1y<14x+3

288.

{x+2y<4y<x2{x+2y<4y<x2

289.

{3xy6y12x{3xy6y12x

290.

{2x+4y8y34x{2x+4y8y34x

291.

{2x5y<103x+4y12{2x5y<103x+4y12

292.

{3x2y6−4x2y>8{3x2y6−4x2y>8

293.

{2x+2y>−4x+3y9{2x+2y>−4x+3y9

294.

{2x+y>−6x+2y−4{2x+y>−6x+2y−4

295.

{x2y<3y1{x2y<3y1

296.

{x3y>4y1{x3y>4y1

297.

{y12x3x2{y12x3x2

298.

{y23x+5x3{y23x+5x3

299.

{y34x2y<2{y34x2y<2

300.

{y12x+3y<1{y12x+3y<1

301.

{3x4y<8x<1{3x4y<8x<1

302.

{−3x+5y>10x>−1{−3x+5y>10x>−1

303.

{x3y2{x3y2

304.

{x1y3{x1y3

305.

{2x+4y>4y12x2{2x+4y>4y12x2

306.

{x3y6y>13x+1{x3y6y>13x+1

307.

{−2x+6y<06y>2x+4{−2x+6y<06y>2x+4

308.

{−3x+6y>124y2x4{−3x+6y>124y2x4

309.

{y−3x+23x+y>5{y−3x+23x+y>5

310.

{y12x1−2x+4y4{y12x1−2x+4y4

311.

{y14x2x+4y<6{y14x2x+4y<6

312.

{y3x1−3x+y>−4{y3x1−3x+y>−4

313.

{3y>x+2−2x+6y>8{3y>x+2−2x+6y>8

314.

{y<34x2−3x+4y<7{y<34x2−3x+4y<7

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

315.

Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for $15 and the landscapes for $10. She needs to sell at least $800 worth of drawings in order to earn a profit.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Will she make a profit if she sells 20 portraits and 35 landscapes?
  4. Will she make a profit if she sells 50 portraits and 20 landscapes?
316.

Jake does not want to spend more than $50 on bags of fertilizer and peat moss for his garden. Fertilizer costs $2 a bag and peat moss costs $5 a bag. Jake’s van can hold at most 20 bags.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Can he buy 15 bags of fertilizer and 4 bags of peat moss?
  4. Can he buy 10 bags of fertilizer and 10 bags of peat moss?
317.

Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Can she mail 60 cards and 26 packages?
  4. Can she mail 90 cards and 40 packages?
318.

Juan is studying for his final exams in Chemistry and Algebra. He knows he only has 24 hours to study, and it will take him at least three times as long to study for Algebra than Chemistry.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Can he spend 4 hours on Chemistry and 20 hours on Algebra?
  4. Can he spend 6 hours on Chemistry and 18 hours on Algebra?
319.

Jocelyn is pregnant and needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of $15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost $0.35 each and the granola bars cost $2.50 each.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could she buy 5 bananas and 6 granola bars?
  4. Could she buy 3 bananas and 4 granola bars?
320.

Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein 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 gram. If he has $ 10 dollars to spend

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could he buy 3 bottles of protein water and 1 protein bar?
  4. Could he buy no bottles of protein water and 5 protein bars?
321.

Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?
  4. Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?
322.

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.

Write a system of inequalities to model this situation.
Graph the system.
Could he meet his goal by walking 3 miles and running 1 mile?
Could he meet his goal by walking 2 miles and running 2 mile?

Everyday Math

323.

Tickets for an American Baseball League game for 3 adults and 3 children cost less than $75, while tickets for 2 adults and 4 children cost less than $62.

  1. Write a system of inequalities to model this problem.
  2. Graph the system.
  3. Could the tickets cost $20 for adults and $8 for children?
  4. Could the tickets cost $15 for adults and $5 for children?
324.

Grandpa and Grandma are treating their family to the movies. Matinee tickets cost $4 per child and $4 per adult. Evening tickets cost $6 per child and $8 per adult. They plan on spending no more than $80 on the matinee tickets and no more than $100 on the evening tickets.

  1. Write a system of inequalities to model this situation.
  2. Graph the system.
  3. Could they take 9 children and 4 adults to both shows?
  4. Could they take 8 children and 5 adults to both shows?

Writing Exercises

325.

Graph the inequality xy3xy3. How do you know which side of the line xy=3xy=3 should be shaded?

326.

Graph the system {x+2y6y12x4{x+2y6y12x4. What does the solution mean?

Self Check

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

This table has four columsn and four rows. The columns are labeled, “I can………,” “confidently.” “with some help.” “no – I don’t get it!” The only rows filled in are under the “I can……...” column. The rows say, “determine whether an ordered pair is a solution of a system of linear inequalities.” “solve a system of linear inequalities by graphing.” and “solving applications of systmes of inequalities.”

After reviewing this checklist, what will you do to become confident for all objectives?

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