Absolute Value Equations

In this lesson you’ll learn what absolute value means and how to solve simple absolute value equations.

Absolute value $|x|$ is the distance from 0 on a number line, always positive or zero.

$|x| = 5$ means $x = 5$ or $x = -5$ (two solutions).

Solving $|\text{expression}| = \text{number}$

1. Set up two cases: expression $= \text{number}$ or expression $= -\text{number}$
2. Solve each equation separately.

Example: $|x + 3| = 7$

$$\begin{aligned} x + 3 &= 7 & x + 3 &= {-}7 \\ x &= 4 & x &= {-}10 \end{aligned}$$

Solutions: $x = 4$ or $x = -10$

Check: $|4 + 3| = |7| = 7$ ✓ and $|-10 + 3| = |-7| = 7$ ✓

Note: If $|\text{expression}| = \text{negative number}$ → no solution (absolute value can’t be negative).

Solve $|2x - 4| = 10$

Case 1:

$$\begin{aligned} 2x - 4 &= 10 \\ 2x &= 14 \\ x &= 7 \end{aligned}$$

Case 2:

$$\begin{aligned} 2x - 4 &= {-}10 \\ 2x &= {-}6 \\ x &= {-}3 \end{aligned}$$

Check: $|2(7) - 4| = |14 - 4| = 10$ ✓ and $|2(-3) - 4| = |-6 - 4| = |-10| = 10$ ✓

Absolute value represents distances or differences without direction:

• Temperature 5 degrees from target: $|t - 72| = 5$ → $t = 77$ or $t = 67$
• Budget overrun/underrun: $|\text{spent} - \text{budget}| \leq 20$ dollars
• Error tolerance: $|\text{measurement} - \text{true value}| < 0.05$

These help with tolerances, errors, or ranges in measurements and finance.