Lesson 6-3

Solving Multi-Step Inequalities

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Objectives

• Solve inequalities involving more than one operation

• Solve linear inequalities involving the Distributive Property

Vocabulary

• none

Multiplication and Division PoEProperties of Equality (PoE) are based on the concept

that as long as you do the same thing to both sides of an equation, then you have not changed anything.

• Multiplication PoE– For any numbers a, b, and c, if a = b, then ac = bc

– You can multiply both sides of an equation by the same thing without changing the equation

• Division PoE– For any numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c

– You can divide both sides of an equation by the same thing without changing the equation

• Multiplication and division are reciprocal actions

Example 1

Science The inequality F > 212 represents the

temperatures in degrees Fahrenheit for which water is

a gas (steam). Similarly, the inequality

represents the temperatures in degrees Celsius for

which water is a gas. Find the temperature in degrees

Celsius for which water is a gas.

Example 1 cont

Answer: Water will be a gas for all temperaturesgreater than 100°C.

Original inequality

Subtract 32 from each side.

Simplify.

Multiply each side by

Simplify.

Example 2

Solve 13 – 11d ≥ 79. Then check your solution.

Original inequality

Subtract 13 from each side.

Simplify.

Divide each side by –11 andchange

Simplify.

Check To check the solution, substitute –6, a number

less than –6, and a number greater than –6.

Answer: The solution set is

Example 3Write an inequality for the sentence below. Then solve the inequality.

Four times a number plus twelve is less than a number minus three.

Four timesa number plus

is less than

a numberminus three. twelve

4n + <12

Example 3 cont

Original inequality

Subtract n from each side.

Simplify.

Subtract 12 from each side.

Simplify.

Divide each side by 3.

Simplify.

Answer: The solution set is

Example 4

Original inequality

Add c to each side.Simplify.Subtract 6 from each side.Simplify.

Divide each side by 4.

Simplify.

Combine like terms.Distributive Property

Answer: The solution set is

Example 5

Answer: Since the inequality results in a falsestatement, the solution set is the empty set Ø.

Original inequality

Distributive Property

Combine like terms.

Subtract 4s from each side.

This statement is false.

Summary & Homework

• Summary:– Multi-step inequalities can be solved by undoing

the operations– Remember to reverse the inequality sign when

multiplying or dividing each side by a negative number

– When solving equations that contain grouping symbols, first use the Distributive Property to remove the grouping symbols

• Homework: – Pg 335 16-38 even