Two-sided Inequalities

Contributor: Mason Smith. Lesson ID: 11242

There are two sides to every argument, and often two sides to inequalities. Don't let that throw you; the steps are equally as simple as for simpler problems. Practice online and with word problems!


High School, Middle School

learning style
personality style
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!


Q: What did Miss Manners say to the inequality symbol?

A: It's not polite to point.

Solving inequalities with variables on both sides often seems confusing at first, but that's what you are here to learn!

Let's start with a simple example: x < 3x + 8.

First, bring the x over to the side where it will stay positive if possible; that gives us 0 < 2x + 8.

If we brought the x the other way, then our x would be negative and we would have to flip the sign and the entire process of solving the inequality becomes much more difficult than it needs to be.

Now we have 0 < 2x+ 8, which you can solve, since that is just a multi-step inequality, which we covered in the previous lesson. Give it a try and see if you came up with the same answer and graph as below.

You should have gotten the answer of -4 < x (or x > -4 ) which, when graphed, looks like this:

Try the problem 6x - 1 < 3.5x + 4. Do you think you can solve it without looking at the following instructions? Try it first before following along with the steps below!

When we bring the variable (x) to the side where it will be positive, we are left with

2.5x -1 < 4.

To get the variable by itself, we have to add 1 to both sides so we have 2.5 x < 5.

Finally, we must divide both sides by 2.5 to get x by itself, and we are left with x < 2, which we can easily graph as:

Now that you know you can solve inequalities with variables on both sides similarly to equations, let's kick it up a notch and try a more difficult problem: 6(1-x) < 3x.

First, we must distribute the 6 by multiplying it by 1 and –x to get 6 - 6x < 3x.

Now we will add 6x to both sides to get 6 < 9x.

When we divide by 9, we end up with our final answer of:  6   < x


Often, when we deal with inequalities, it revolves around comparing different plans offered by companies in order to find out which is the most economical or cheapest, and these often require multiple variables to solve. Let's try a business problem together.

The New York Times (NYT) charges $650 plus $80 a week to run an ad. The Daily Prophet charges $145 per week to run the same ad. For how many weeks w is it less expensive to run the ad in the Daily Prophet?

First, let's break down both of the rates. NYT charges 650 plus 80 per week, so their rate will be 650 + 80 w. The Daily Prophet charges $145 per week, so their rate is just 145w where w is the number of weeks.

Now, we need to compare to find out when the Daily Prophet is less than the NYT, so we just use the inequality 145w < 650 + 80w.

When we solve, we end up with 65w < 650, which gives an answer of w < 10. As long as the ad is less than 10 weeks long, the Daily Prophet is cheaper.

Now that we understand how to tackle inequalities with variables on both sides, let's practice solving them.

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