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Solving Inequalities in One Variable



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:




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

  • 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.


Inequality symbols


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.



Inequality graph



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:


Rules for Solving Inequalities

  • 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.



Example 1


solving inequalities




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.




Example 2


inequalities examples




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:


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






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.



See What Happens.....


solving an algebra inequality



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.



Example 3


solving 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.


Example 4


solving inequalities with fractions



Now are you ready for a few on your own?

solving inequality practice problems

Click here to move onto Solving Inequalities Practice Problems.







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