Rules for Operations on Inequalities

By Mary Jane Sterling

If your finite math instructor asks you to solve a linear inequality, you can use most of the same rules that you’d use when solving linear equations. There are two huge exceptions, however, which you’ll learn about here.

The following list shows all the rules you need to know when performing operations on inequalities. Note that although only the < symbol is shown in this list, these same rules apply to any inequality, including >, ≤, and ≥.

  • If a < b, then a + c < b + c. Adding the same number to each side of an inequality does not change the direction of the inequality symbol.
  • If a < b, then ac < bc. Subtracting the same number from each side of an inequality does not change the direction of the inequality symbol.
  • If a < b and if c is a positive number, then a · c < b · c.
    Multiplying each side of an inequality by a positive number does not change the direction of the inequality symbol.
  • If a < b and if c is a positive number, then
    FNTMATH_0602
    Dividing each side of an inequality by a positive number does not change the direction of the inequality symbol.
  • If a < b and if c is a negative number, then a · c > b · c.
    Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol.
  • If a < b and if c is a negative number, then
    FNTMath-0603
    Dividing each side of an inequality by a negative number reverses the direction of the inequality symbol.

Now, let’s apply those rules to some examples. First, simplify the linear inequality 4x – 3 ≥ 21 and solve for x. You first need to add 3 to each side, and then divide each side by 4. The inequality symbol remains in the same direction.

FNTMATH_0604

Any number 6 or greater is a solution of the inequality 4x – 3 ≥ 21.

Now let’s try an example that involves dividing by a negative number: solve 16 – 5x < 11 for x. In this case, you first need to subtract 16 from each side and then divide by –5. Dividing by a negative number means you reverse the inequality symbol.

FNTMATH_0605

Any number greater than 1 is a solution of the inequality 16 – 5x < 11.