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Mom's Basement University

Absolute Value Equations

What You’ll Learn

Section titled “What You’ll Learn”

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

The Concept

Section titled “The Concept”

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

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

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

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

Example: ∣x+3∣=7|x + 3| = 7

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

Solutions: x=4x = 4 or x=−10x = -10

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

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

Worked Examples

Section titled “Worked Examples”

Solve ∣2x−4∣=10|2x - 4| = 10

Case 1:

2x−4=102x=14x=7\begin{aligned} 2x - 4 &= 10 \\ 2x &= 14 \\ x &= 7 \end{aligned}

Case 2:

2x−4=−102x=−6x=−3\begin{aligned} 2x - 4 &= {-}10 \\ 2x &= {-}6 \\ x &= {-}3 \end{aligned}

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

Real-World Applications

Section titled “Real-World Applications”

Absolute value represents distances or differences without direction:

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

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

Quiz

Section titled “Quiz”
Solve $|x| = 6$.
Solve $|x - 5| = 3$.
Solve $|2x + 1| = 9$.
A temperature is within 4 degrees of 70°F. Which equation?
Solve $|3x| = 15$.