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.
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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