Absolute Value Equations and Inequalities: Concepts, Methods, and Applications

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:

$|X| = a \implies X = a \text{ or } X = -a$

Example: Solve |x| = 5.

• Solutions: x = 5 or x = -5

Worked Example

Solve 8|x − 3| = 4

• Divide both sides by 8: $|x-3| = \frac{1}{2}$

• Apply the definition: $x-3 = \frac{1}{2}$ or $x-3 = -\frac{1}{2}$

• So, $x = 3 + \frac{1}{2} = \frac{7}{2}$ or $x = 3 - \frac{1}{2} = \frac{5}{2}$

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: $|X| < a$ is equivalent to $-a < X < a$

• Type 2: $|X| > a$ is equivalent to $X < -a$ or $X > a$

For a > 0 and an algebraic expression X:

• $|X| < a \iff -a < X < a$

• $|X| > a \iff X < -a \text{ or } X > a$

Example 1: $|4x| < 12$

• By definition: $-12 < 4x < 12$

• Divide by 4: $-3 < x < 3$

• Interval Notation: (−3, 3)

Example 2: $|5x + 1| < 8$

• By definition: $-8 < 5x + 1 < 8$

• Subtract 1: $-9 < 5x < 7$

• Divide by 5: $-\frac{9}{5} < x < \frac{7}{5}$

• Interval Notation: $\left[ -\frac{9}{5}, \frac{7}{5} \right]$

Example 3: $|5x + 1| > 8$

• By definition: $5x + 1 < -8$ or $5x + 1 > 8$

• Solve each inequality separately:

  • $5x + 1 < -8 \implies 5x < -9 \implies x < -\frac{9}{5}$

  • $5x + 1 > 8 \implies 5x > 7 \implies x > \frac{7}{5}$

• Interval Notation: $( -\infty, -\frac{9}{5} ) \cup ( \frac{7}{5}, \infty )$

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:

• $|T - 70| \leq 4$

• So, $66 \leq T \leq 74$

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:

• $|L - 10| \leq 0.03$

• So, $9.97 \leq L \leq 10.03$