# Solving Linear Inequalities

**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

**Graphing Inequalities on a
Number Line & Interval Notation**

**Solution Set
**• Solution set – all values that satisfy an

inequality

• Often a solution set is expressed using a

number line

**Graphing Inequalities on a Number
Line
**• Consider the inequality x > 1

–What are some values of x that make the

inequality true?

• {2, 3, 4, 5, …}

– 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

• 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)

**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 ∞

• 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 < -3

b) y ≥ 9⁄2

**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

**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?

• 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

**Solving Inequalities Using
Both Properties**

**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)

Ex 5: Solve, graph, AND write the solution

set in interval notation:

**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)