Simplify Absolute Value Expressions


This is a tutorial on how to simplify expressions with absolute value. It is important to understand the definition of the absolute value below first. Expressions with absolute value can be simplified only when the sign of the expression inside the absolute value is known

If x > = 0 then | x | = x and if x < 0 then | x | = -x.

Follow the examples below and try to understand each step.

Example 1: Simplify the expressions with absolute value

  1. | -2 + 10 |

  2. | 1/2 -20 |

  3. | sqrt(3) - 5 |

  4. | sqrt(14) - 3*pi + 10 |

Solution to Example1

  1. - 2 + 10 = 8 is positive and according to the definition above | -2 + 10 | can be simplified as follows
    | -2 + 10 | = | 8 | = 8

  2. 1/2 - 20 = -39/2 is negative, according to the definition above | 1/2 -20 | can be simplified as follows
    | 1/2 -20 | = | -39/2 | = -(-39/2) = 39/2

  3. sqrt(3) - 5 is approximately equal to -3.27 which is negative, according to the definition above | sqrt(3) - 5 | can be simplified as follows
    | sqrt(3) - 5 | = - ( sqrt3 - 5) = 5 - sqrt(3)

  4. sqrt(14) - 3*pi + 10 is approximately equal to 4.32 which is positive, according to the definition above | sqrt(14) - 3*pi + 10 | can be simplified as follows
    | sqrt(14) - 3*pi + 10 | = sqrt(14) - 3*pi + 10


Examples with algebraic expressions are now presented.

Example 2: Simplify the algebraic expressions with absolute value

  1. | x2 + 1 |

  2. | x + 3| , if x < -3

  3. | -x + 2 | , if x > 2

Solution to Example 2

  1. x2 + 1 is always positive and according to the definition above | x2 + 1 | can be simplified as follows
    | x2 + 1 | = x2 + 1

  2. if x < -3 then x + 3 < 0. According to the definition above | x + 3 | can be simplified as follows
    | x + 3 | = -(x + 3) = -x - 3

  3. if x > 2 then x - 2 > 0 and -x + 2 < 0. According to the definition above | -x + 2 | can be simplified as follows
    | -x + 2 | = - ( - x + 2 ) = x - 2

More links and references to absolute value in functions, equations, problems, tutorials and self tests.


Definition of the absolute value.

Absolute value Functions.

Solve Equations with Absolute Value.

Absolute Value Equations And Inequalities Problems.

Tutorial on Absolute Value Inequalities.