How to Do Inequalities in Mathematics

Adding and Subtracting Values

• 1

  Examine the inequality to determine whether to add, or subtract, a value from both sides of the equation. This is called the Addition/Subtraction Property for Inequalities. (If a < b, then a + c < b + c, and if a < b, a - c < b -- c.)

• 2

  Perform the addition or subtraction. For example, in the inequality X + 5 < 12, subtract 5 from both sides. X + 5 -- 5 < 12 -- 5. The solution to the inequality is X < 7(any value less than 7). For another example, 15 < X - 5, add 5 to both sides. 15 + 5 < X -- 5 + 5. The solution to the inequality is 20 < X.

• 3

  Reverse the sides of the inequality. Traditionally, the X, or other variable, is on the left side of the inequality. X > 20. Remember to reverse the inequality symbol.

Multiplying or Dividing by a Value

• 4

  Determine whether to multiply, or divide, by a value on both sides of the inequality. If the value used to multiply or divide is a positive number, use the Multiplication/Division Properties for Inequalities when multiplying or dividing with positive values. (If a < b, and c is positive, then ac < bc, and if a < b, and c is positive, a/c < b/c.)

• 5

  Perform the multiplication or division. For example, in the inequality 5X > 15, divide both sides by 5. 5X ÷ 5 > 15 ÷ 5. The solution to the inequality is X > 10. For another example, 5X/10 > 20, multiply both sides by 2. 5X/10 x 2 > 20 x 2. The solution to the inequality is X > 10.

• 6

  Use the Multiplication/Division Properties for Inequalities when multiplying or dividing with negative numbers. (If a < b, and c is negative, ac > bc, and if a < b, and c is negative, then a/c > b/c.) A negative value, multiplied or divided on both sides of the inequality, reverses the sign of the inequality. For example, -X/3 > 7, multiply both sides by. (-X/3) x (-3) < (7) x (-3), and reverse the inequality sign immediately. The solution of the inequality is X < -21. Remember that multiplying two negative values results in a positive value, such as (-X/3) x (-3) = X. For another example, -2X < -12, divide both sides by -2. -2X/-2 > -12/-2 (reverse the inequality sign immediately). The solution to the inequality is X > 6.