Lesson 3.6 – 3.7

Solve One-Step Inequalities

Essential Question: How do you solve an inequality?

Before we start…

This month there are a lot of movies coming out that you want to see. You have $45 to spend and each ticket to the movie costs $7. What is the maximum number of movies you can see this month?

What is an inequality?

A statement formed by placing an inequality symbol between two expressions

𝑥≤6

Symbol Meaning

Is less than

Is less than or equal to

Is greater than

Is greater than or equal to

What is a solution of an inequality?

The set of numbers that you can substitute for the variable to make the inequality true

What is the graph of an inequality?

Consists of all points on a number line that represent solutions.

How do you graph an inequality?

• You will make a circle on the number given.– If it’s equal to, you will fill it in– If it’s not equal to, you will leave it open

• Then draw an arrow from the circle in the direction of the inequality– To the right if it’s greater than– To the left if it’s less than

Examples

Graph

Graph

2x

1x

Examples

Graph

Graph

5x

7x

What are equivalent inequalities?

Inequalities that have the same solutions as the original inequality.

How do you solve an inequality?

• You will use inverse operations to isolate the variable.

• If you multiply or divide by a negative number then you must flip the inequality symbol.

• Once solved, you’ll graph the solution set on a number line.

• Open circle if it’s greater than or less than.• Closed circle if it’s greater than or equal to or less than

or equal to.

Transformations That Produce Equivalent Inequalities

Transformation Applied Original Inequality Equivalent Inequality

Add the same number to each side.

Subtract the same number from each side.

Multiply each side by the same positive number.

Divide each side by the same positive number.

Multiply each side by the same negative number and reverse the inequality.

Divide each side by the same negative number and reverse the inequality.

47 x 11x

13 x 2x10

2

1x 20x

155 x 3x

17 x 17x

459 x 5x

Solve and graph

4 5x

Solve and graph

54

x

Solve and graph

9 3x

Solve and graph

36

x

Solve and graph

3 24x

Solve and graph

7 4x

Solve and graph

5 45v

Translating to Equations and Inequalities

Sometimes two inequalities are combined.

Luggage Weights You are checking a bag at an airport. Bags can weigh no more than 50 pounds. Your bag weighs 16.8 pounds. Find the possible weights w (in pounds) that you can add to the bag.

Internet You earn points from buying items at an Internet shopping site. You would like to redeem 2350 points to get an item for free, but you want to be sure to have no more than 6000 points left over. What are the possible number of points you can have before making a redemption?

Music You have $90 to buy CDs for your friend’s party. The CDs cost $18 each. What are the possible numbers of CDs that you can buy?

Apples An empty basket weighs 2 pounds. When filled with apples, the basket weighs more than 13 pounds. Write and solve an inequality that represents the situation. What does the variable represent?

Calling Card You have $9.50 with which to recharge your phone card. Write and solve an inequality to find how many minutes you can add to your phone card if each minute costs $0.10.

How do you solve an inequality?

Ticket Out the Door

Solve and graph the inequality:

6 24x