3.15: Solve Inequalities Using Division
“I think I need new band shoes,” Kayla said to her Mom at breakfast one morning.
“Why is that? Have your feet grown?” her Mom asked taking a sip of coffee.
“I’m not sure, but I know that I have a blister on my toe from them.”
“Alright, must be time for new shoes. You don’t have band practice today, so let’s go after school and get new shoes. That will give you time to break them in before Friday’s game.”
After school, Kayla’s Mom picked her up and they went to the shoe store. Kayla first picked out a very fancy pair of shoes, but they weren’t on sale. Her Mom picked out a nice pair that would work fine, but were a little more economically priced.
“These are halfoff. Let’s get them. They will work fine,” her Mom said.
Kayla agreed and they purchased the shoes. Kayla saw her Mom hand over $40.00 and get some change back.
The shoes that Kayla and her Mom bought were half off the original price. They spend less than $40.00 on the shoes. Can you figure out three possible original prices for Kayla’s shoes?
Figuring this out will require you to use inequalities with multiplication and division. Take your time while working on this. At the end of the Concept, you will see this problem again.
Guidance
An inequality is a mathematical statement where a quantity may be equal to or less than or greater than another quantity. You can identify an inequality and solve inequalities using all four operations. Now we are going to look at solving inequalities that involve multiplication.
Let’s look at this more closely.
Just as you sometimes need to multiply or divide both sides of an equation by the same number in order to solve an equation, you may sometimes need to multiply or divide both sides of an inequality by the same number in order to solve it. However, doing so may be a little more complicated.
Do you know how to divide and solve inequalities?
One of the first things you may be wondering is what happens if you divide both sides of an inequality by the same number.
Remember, dividing by a number,
The division property of inequality states that if each side of an inequality is divided by the same positive number, the sense of the inequality stays the same. In other words, the inequality symbol does not change.
If
If
However, if each side of an inequality is divided by the same negative number, the sense of the inequality changes and the inequality symbol must be reversed.
If
If
Remember, that to solve an inequality by using division, you will need to see multiplication in the problem. This is because the inverse of multiplication is division and we use inverse operations to solve inequalities.
Solve this inequality:
Isolate the variable by using inverse operations. Since 3 is multiplied by
The answer is the
Identify the inequality symbol that goes in each blank.
Example A
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
First, write an inequality to describe the values in the problem.
Since Kayla’s Mom gave the clerk $40.00 and received change, we know that the shoes were less than $40.00.
Here is our inequality.
Now we can solve it for possible prices.
Three possible original prices could be $75.00, $60.00 or $55.00. But there are many possible options.
Vocabulary
 Inequality
 a mathematical statement where two quantities may or may not be equal. There are many possible answers to an inequality.
 Multiplication Property of Inequality
 multiplication of a positive number = inequality is the same. Multiplication of a negative number, the sense of the inequality changes and the sign is reversed.
 Division Property of Inequality
 Division by a positive number = inequality is the same. Division by a negative number, the sense of the inequality changes and the sign is reversed.
Guided Practice
Here is one for you to try on your own.
Maddie is buying bottles of water for a school event. The store only sells bottles of water in sixpacks. That is, bottles of water are only sold in packages of 6 bottles each. Maddie wants to make sure there are at least 72 bottles of water available at the event.
a. Write an inequality to represent,
b. If Maddie buys 11 sixpacks of water for the event, will that be enough? Explain.
Solution
Consider part a first.
Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. Since bottles of water are sold in packages of 6, or sixpacks, you can find the total number of bottles purchased by multiplying 6 by the number of packages purchased. The key words “at least” indicate that you should use a
The equation
You may also want to consider that the value of
The reason that the value of
Next, consider part b.
Solve the inequality to help you.
Now, let's consider if Maddie could buy 11 sixpacks of water and have enough for the event. According to the inequality above, the number of sixpacks,
That means that if Maddie bought only 11 sixpacks of bottled water, she would not have enough water for the event.
Video Review
Khan Academy Inequalities Using Multiplication and Division
Practice
Directions: Solve each inequality using division.

3x<12 
5x>60 
6x<72 
9x>81 
11x>121 
12x≤48 
13x≥39 
15x≤60 
22x>66 
48x≤192
Directions: Solve each inequality.

4x>−36 
−9n≥63 
7n≥63 
−19x≤38 
−9y<18
Division Property of Inequality
The division property of inequality states that two unequal values divided by a positive number retain the same relationship. Two unequal values divided by a negative number result in a reversal of the relationship.Equivalent Fractions
Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.inequality
An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .Multiplication Property of Inequality
The multiplication property of inequality states that when an inequality is multiplied by a negative number, the sense of the inequality changes and the sign is reversed.Image Attributions
Description
Learning Objectives
Here you'll solve inequalities using division.