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Graphing Systems of Linear Equations and Inequalities Study Guide (page 3)

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Updated on Aug 24, 2011

Solving Systems of Inequalities Graphically

A system of inequalities is two or more inequalities with the same variables. You graph systems of linear inequalities like you graph systems of linear equations. However, remember that the graph of an inequality consists of a boundary line and a shaded area. Review Lesson 10 if you need help recalling the basics of graphing inequalities.

To graph a system of inequalities, transform each inequality into y = mx + b, and graph the boundary line. Then determine if you will shade above or below the boundary line. The solution of the system of inequalities will be the intersection of the shaded areas.

Example:

y > x

y < 3

The inequalities are already in y = form, so you are ready to graph them. The slope of the first equation is 1, and the y-intercept is 0. Start with the y-intercept of 0 and then go up 1 and to the right 1. The boundary line will be dotted because the inequality symbol is >. Because the inequality is >, you will shade above the line.

Graphing Systems of Linear Equations and Inequalities

The inequality y < 3 will have a horizontal boundary line. The boundary line will be dotted, and you will shade below the line because the inequality symbol is <.

Graphing Systems of Linear Equations and Inequalities

The intersection of the two shaded areas is the solution of the system.

Graphing Systems of Linear Equations and Inequalities

Example:

–x + y ≤ 4

2x + y ≥ 1

Transform the first inequality to y = mx + b. –x + y ≥ 4
Add x to both sides ofthe inequality. –x + x + y ≥ 4 + x
Simplify. y ≥ 4 + x
Use the commutative property. y ≥ x + 4
Transform the second inequality to y = mx + b. 2x + y ≤ 1
Subtract 2x from both sides ofthe inequality 2x – 2x + y ≤ 1 – 2x
Simplify. y ≤ 1 – 2x
Use the commutative property. y ≤ –2x + 1

 

The slope of the first inequality is 1 and the y-intercept is 4. To graph the inequality, start with the y-intercept, which is 4. From that point, go up 1 and to the right 1. Draw a line through the starting point and the endpoint. The boundary line will be a solid line, and you will shade above it because the inequality symbol is ≥.

Graphing Systems of Linear Equations and Inequalities

The slope of the second inequality is –2, and the y-intercept is 1. To graph the inequality, start with the y-intercept, which is 1. From that point, go down 2 and to the right 1. Draw a line through the starting point and the endpoint. The boundary line will be a solid line, and you will shade below the line because the inequality symbol is ≤.

Graphing Systems of Linear Equations and Inequalities

The solution of the system of inequalities is the intersection of the two shaded areas.

Graphing Systems of Linear Equations and Inequalities

Skill Building until Next Time

Use a system of inequalities to represent how you will spend your money. Let x = the amount of money you need to spend on necessities. Let y = the amount of money you can spend on recreation. Fill in the system of inequalities with the amount of money that fits your circumstances.

x + y ≤ (amount of money you have to spend)

x ≥ (amount of money you need to spend on necessities)

When you have filled in the dollar amounts, graph the system of inequalities. You do not want any values less than zero. Why?

Find practice problems and solutions for these concepts at Graphing Systems of Linear Equations and Inequalities Practice Problems.

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