How to Solve Inequalities With Negative Numbers

Linear inequalities are equations with a less-than, "<", or greater-than sign, ">," instead of an equal sign. Solve these equations by applying the same algebraic rules as if you have equal signs. Just remember, whenever you divide or multiply both sides of the inequality equation by a negative number, you have to change the sign. In this case, a less-than sign changes to a greater-than sign, and a greater-than sign changes to a less-than sign. This ensures you maintain the proper relationship between the two sides.

Instructions

    • 1

      Write out the inequality equation clearly. For example:

      3x + 6 > 8x + 4

    • 2

      Solve the equation to combine like terms. Follow the order of operations regarding how to manipulate and combine terms. You can easily remember the order of operations with the acronym PEMDAS, which stands for parentheses, exponents, multiplication and division, and then addition and subtraction. Using the example, first subtract 8x from both sides to get the x terms on the left side of the equation:

      3x + 6 - 8x > 8x + 4 - 8x

      Which leaves -5x + 6 > 4.

      Next, subtract 6 from both sides to get the constant terms on the right side of the inequality sign:

      -5x + 6 - 6 > 4 - 6

      Which leaves -5x > -2.

    • 3

      Manipulate the problem to get x on one side of the inequality sign by itself. Again, adhere to PEMDAS. Continuing with the example, divide both sides by 5 to get x on the left side by itself:

      -5x/5 > -2/5

      Which leaves -x > -2/5.

      Because you have -x, multiply both sides by -1 to convert -x into x:

      -x (-1) > -2/5 (-1)

      Which leaves x < 2/5. Note that the sign switched from greater-than to less-than, because whenever you multiply or divide both sides of the inequality by a negative, the sign changes.

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