Solving Linear Inequalities

MATH 018

Combined Algebra

S. Rook

2

Overview

• Section 2.7 in the textbook:– Graphing inequalities on a number line &

interval notation– Using the Addition Property of Inequality– Using the Multiplication Property of Inequality– Solving inequalities using both properties

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Graphing Inequalities on a Number Line & Interval Notation

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Solution Set

• Most equations have 1 solution whereas most inequalities have an infinite number of solutions

• Solution set – all values that satisfy an inequality

• Often a solution set is expressed using a number line

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Graphing Inequalities on a Number Line

• Consider the inequality x > 1– What are some values of x that make the

inequality true?– Thus x can be any value greater than 1

• Which direction on the number line indicates increasing values?

– Can x = 1 be a solution to the inequality?• Since x = 1 is not in the solution set, we put a

parenthesis around 1 on the number line

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Graphing Inequalities on a Number Line (Continued)

• Now consider x ≥ 1– Graphed ALMOST the same way EXCEPT:

• Is x = 1 included in the solution set?– Since x = 1 is part of the solution set, we put a

bracket around 1 on the number line

• Given x < a or x > a:– Parenthesis goes around a on the number line (not

inclusive)

• Given x ≤ a or x ≥ a:– Bracket goes around a on the number line (inclusive)

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Interval Notation

• Easy once the graph is obtained

• Represents the “endpoints” of the graph– First “value” is what is shaded furthest to the

left on the graph– Second “value” is what is shaded furthest to

the right on the graph– A shaded arrow represents oo

• Parentheses ALWAYS go around infinity

Graphing Inequalities on a Number Line & Interval Notation (Example)

Ex 1: Graph on a number line AND write in interval notation:

a) x < 9⁄2b) -5 < y ≤ 2

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Addition Property of Inequality

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Addition Property of Inequality

• Works the same way as the Addition Property of EQuality

• When a number is being added or subtracted to the variable:– Add the OPPOSITE number to BOTH SIDES

• Consider 2 < 7– What happens when we add 2 to both sides?– What happens when we subtract 5?

Addition Property of Inequality (Example)

Ex 2: Solve, graph, AND write the solution set in interval notation:

a) x + 3 > 4

b) 3y – 4 ≤ 2y – 9

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Multiplication Property of Inequality

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Multiplication Property of Inequality

• ALMOST the same as the Multiplication Property of EQuality

• When the variable is being multiplied by a number:– Divide BOTH SIDES by the number

INCLUDING THE SIGN

• Consider 4 > 2– What happens when we divide by 2?– What happens when we divide by -2?

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Multiplication Property of Inequality (Continued)

• Thus, when we DIVIDE an INEQUALITY by a NEGATIVE number:– Switch the direction of the inequality– Failing to SWITCH the inequality when

DIVIDING by a NEGATIVE number is a common mistake!

Multiplication Property of Inequality (Example)

Ex 3: Solve, graph, AND write the solution set in interval notation:

a) 6x ≥ -18

b) -8y < 14

c) -z ≤ -6

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Solving Inequalities Using Both Properties

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Solving Inequalities Using Both Properties (Example)

Ex 4: Solve, graph, AND write the solution set in interval notation:

a) -5x – 9 > -2x + 12

b) 4(y – 5) ≥ 3y – (y + 2)

Solving Inequalities Using Both Properties (Example)

Ex 5: Solve, graph, AND write the solution set in interval notation:

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4

1

2

321

4

3

xx

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Summary

• After studying these slides, you should know how to do the following:– Graph an inequality on a number line– Understand the Addition and Multiplication Properties

of Inequality– Solve and graph inequalities

• Additional Practice– See the list of suggested problems for 2.7

• Next lesson– Solving Absolute Value Equations (Section E.1)