Absolute Value Equations and Inequalities

Definition of Absolute Value

The absolute value of a real number is its distance from zero on the number line, regardless of direction. It is denoted by |x|.

• Key Property: For any real number x, |x| ≥ 0.

• Examples: |5| = 5; |−5| = 5

Solving Absolute Value Equations

To solve equations involving absolute value, follow these steps:

• Step 1: Isolate the absolute value expression if necessary.

• Step 2: Apply the definition of absolute value: For |X| = a (where a ≥ 0), the solutions are X = a or X = −a.

General Form: For a > 0 and an algebraic expression X:

Example: Solve |x| = 5.

• Solutions: x = 5 or x = -5

Worked Example

Solve 8|x − 3| = 4

• Divide both sides by 8:

• Apply the definition: or

• So, or

Special Cases

If the equation is of the form |X| = b, where b < 0, there is no solution because absolute value cannot be negative.

Solving Absolute Value Inequalities

Absolute value inequalities can be classified into two main types:

• Type 1: is equivalent to

• Type 2: is equivalent to or

For a > 0 and an algebraic expression X:

Example 1:

• By definition:

• Divide by 4:

• Interval Notation: (−3, 3)

Example 2:

• By definition:

• Subtract 1:

• Divide by 5:

• Interval Notation:

Example 3:

• By definition: or

• Solve each inequality separately:

• Interval Notation:

Applications of Absolute Value Inequalities

Example 1: Temperature Range

Suppose you want your room temperature to stay within 4 degrees of 70°F. Let T be the temperature. The requirement can be written as:

• So,

Example 2: Manufacturing Tolerances

A machine is designed to produce metal rods that are 10 cm long, with a maximum allowable error of 0.03 cm. Let L be the length of a rod. The acceptable lengths satisfy:

• So,