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Home » Inequalities » Solving Inequalities

# Solving Inequalities in One Variable

## x < 5 (Read as x is less than 5)

## Rules for Solving Inequalities

## Example 1

## Example 2

### Whenever you **mulitply or divide by a negative number**, you must **reverse the sign**!

## See What Happens.....

## Example 3

## Example 4

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## Other Inequality Lessons That You Might Like

Are you ready to dive into our solving inequalities unit? Let's do a very quick review of inequality basics that you probably first learned about in second grade.

Always remember that inequalities do not have just one solution. There are always multiple solutions! Think about the following inequality:

- We could replace x with 4 because 4 is less than 5.
- We could replace x with 2 because 2 is less than 5.
- We could even replace x with -3 because -3 is less than 5.

We could go on forever, so as you can see there are many many solutions to this inequality!

Let's take a look at the inequality symbols and their meanings again.

When you graph inequalities that have only one variable, we use a number line. We will use open and closed circles and arrows pointing to the left or right to graph our answers.

An open circle on the graph indicates less than (<) or greater than (>).

A closed circle on the graph represents less than or equal to (__<__) or greater than or equal to (__>__).

Take a look at the model below.

Ok... enough review. Let's solve a few inequalities. You are going to solve inequalites using the exact same rules that you used when solving equations. Let's quickly review those rules:

- Whatever you do to one side of the inequality, you must do to the other side.
- Always get rid of the constant first and then any coefficients last.
- If you have fractions, get rid of the fractions first and then proceed with solving the inequality.

There is however, one small rule that you always have to remember when solving inequalities. (Yes... it's always something, isn't it?) We'll get to that in a minute. Let's look at a simple inequality first.

Did you notice that you took the exact same steps that you would've taken in order to solve a regular equation? The only difference was that you had a less than sign instead of an equal sign.

Now that was pretty easy!

The next example is similar to example 1, but I would like to show you how to reverse your answer to make it easier to read and graph. Don't be afraid to do this if your variable ends up the right hand side of the inequality.

Ok... The first two examples should have been pretty easy since you are a superstar at solving equations. Luckily there's only one trick that you have to remember when solving inequalities and that is:

I knew you were going to ask "Why?" And.... you should be asking why. It's important to understand these rules. Sometimes, the best explanations are through examples. Let's take a look.

This works with any true statement as long as you multiply or divide by a negative number. Go ahead, try another one. Write a true statement, and then divide by -2.

Ok... let's look at a few examples.

Our next example revists how to solve equations and/or inequalities with variables on both sides.

Our last example revists how to solve equations and/or inequalities with fractions. I hope you remember the trick.

Now are you ready for a few on your own?

Click here to move onto Solving Inequalities Practice Problems.

- Solving Inequalities Practice Problems
- Solving Inequalities in One Variable

Word Problems - Solving Inequalities Word Problems

Practice Problems - Graphing Linear Inequalities
- Graphing Inequalities Practice Problems
- Solving Systems of Inequalities
- Graphing Systems of Inequalities

Practice Problems - Systems of Inequalities Word Problems
- Systems of Inequalities Word Problems

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