How to Solve Inequalities With Fractions

••• isolated colorful fraction circles image by davidcrehner from

Here is a step-by-step guide to how to solve an inequality with a fraction in it. Even if fractions seem to trip you up every time, once you learn this concept, you'll be solving problems with fractions in them in no time.

    Begin by simply taking in the inequality before you even begin to use any processes to try and solve the problem. Take note of any negatives that you will need to remember to carry through while solving the problem. You should also notice all the processes in the inequality such as multiplication, subtraction, exponents, parentheses and such.

    Use the order of operation in reverse to begin to solve the problem. One easy way to remember the order of operations is to remember the word PEMDAS (parentheses, exponents, multiplication/division, addition/subtraction). Now, when you are solving for a variable, you will use the order of operations in reverse, so instead of beginning with parentheses and ending with addition/subtraction, you will begin with addition/subtraction and end with parentheses.


    If you have the inequality 3<(x/9)+7

    Begin with subtraction by subtracting 7 from both sides, rather than beginning with the parentheses x/9.

    Do all processes to both sides of the inequality until you have solved for x.

    Example: As mentioned in the previous step, you would begin by subtracting 7 from both sides.

    So 3<(x/9)+7 becomes, -4<x/9

    Now you would multiply both sides by 9 because the fraction x/9 is the same as x divided by 9, and the opposite of division is of course multiplication.

    This process leaves you with the solution, -36<x, so x is greater than -36.

    Remember that if your problem requires you to multiply or divide by a negative number, then you need to flip the inequality sign when you do so.

    For example: If instead of multiplying by 9 in the previous problem you had to multiply by -9, you would get 36>x rather than 36<x.


    • Always double check your work.

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.

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  • isolated colorful fraction circles image by davidcrehner from