How to Solve Double Inequalities

By Ainsley Patterson; Updated April 24, 2017
Solve fractional inequalities with a graphing calculator.

Double inequalities may appear too intimidating at first to solve because there are three sides to the equation, but, if you follow the step-by-step guide that provided below you may find them a little less intimidating and a lot easier to solve.

Solving Double Inequalities

Begin by just taking a look at your double inequality before you actually start doing any mathematical processes to the double inequality.

Start solving your double inequality for x by doing all processes to all three parts of the equation. So, just like you would do all processes to both sides of the equation when solving for x with a "regular" equation, you need to do all processes to all sides of the double inequality. For example, if you had the following double equality, 3<2x+8<20, then you would need to do all processes that you do to the middle to both the left and the right as well. For the following steps I will guide you through solving this particular double inequality.
Remember: When solving any kind of equation for a value of x you need to follow the order of operations in reverse, which means that you need to do the processes in the following order: subtraction/addition, multiplication/division, exponents, parentheses. One easy way to remember the order of operations is by remembering the word PEMDAS, Parentheses, Exponents, Multiplication/Division (these two operations are interchangeable), Addition/Subtraction (these two operations are also interchangeable). Now when you are solving an equation, or in this case, a double inequality, for x, simply follow PEMDAS backwards.

Subtract eight from all three sides of the equation. This is what you should be left with when you start with the double inequality 3<2x+8<20: -5<2x<12

Divide all sides of the inequality by two. This is the solution to your double inequality: -2.5<x<6

Remember that if you have to divide or multiply by a negative number in order to get your solution that you need to flip both inequality symbols. If you forget to flip the inequality symbols when multiplying or dividing by a negative number you will not only have the wrong answer, you will have an impossible answer. For example: 3<-2x+8<20 -5<-2x<12 2.5<x<-6 There is no way that x can be greater than 2.5 and less than -6. If you remember to flip the inequality symbols then you get the right solution 2.5>x>-6.

About the Author

Based in Ypsilanti, Mich., Ainsley Patterson has been a freelance writer since 2007. Her articles appear on various websites. She especially enjoys utilizing her more than 10 years of craft and sewing experience to write tutorials. Patterson is working on her bachelor's degree in liberal arts at the University of Michigan.