Absolute Value Equations

What is Absolute Value?

Absolute value is the distance a number is from zero on the number line. Distance is always positive or zero, never negative.

The absolute value of a number x is written as |x|.

Examples:

• |5| = 5 (5 is 5 units from zero)
• |-5| = 5 (-5 is also 5 units from zero)
• |0| = 0 (0 is 0 units from zero)
• |-3.7| = 3.7
• |2 - 7| = |-5| = 5

Key Property of Absolute Value

The absolute value makes all numbers non-negative:

• If x ≥ 0, then |x| = x
• If x < 0, then |x| = -x

Example: |-8| = -(-8) = 8

What is an Absolute Value Equation?

An absolute value equation is an equation that contains an absolute value expression.

Examples:

• |x| = 5
• |x - 3| = 7
• |2x + 1| = 9
• 2|x - 4| + 3 = 11

Why Two Solutions?

Most absolute value equations have TWO solutions because two different numbers can have the same absolute value.

Example: |x| = 5 Both x = 5 and x = -5 work because:

• |5| = 5 ✓
• |-5| = 5 ✓

Think about it: What numbers are 5 units away from zero? Both 5 and -5!

Solving Basic Absolute Value Equations

Form: |x| = a (where a ≥ 0)

Solution: x = a or x = -a

Example 1: |x| = 7 Solution: x = 7 or x = -7

Check:

• |7| = 7 ✓
• |-7| = 7 ✓

Example 2: |x| = 12 Solution: x = 12 or x = -12

Example 3: |x| = 0 Solution: x = 0 (only one solution when a = 0)

Solving |ax + b| = c

Steps:

1. Isolate the absolute value expression
2. Set up two equations:
   • ax + b = c (positive case)
   • ax + b = -c (negative case)
3. Solve both equations
4. Check both solutions

Example 1: |x + 3| = 8

Step 1: Already isolated

Step 2: Set up two equations Case 1: x + 3 = 8 Case 2: x + 3 = -8

Step 3: Solve both Case 1: x = 5 Case 2: x = -11

Step 4: Check

• |5 + 3| = |8| = 8 ✓
• |-11 + 3| = |-8| = 8 ✓

Solution: x = 5 or x = -11

Example 2: |2x - 1| = 7

Case 1: 2x - 1 = 7 2x = 8 x = 4

Case 2: 2x - 1 = -7 2x = -6 x = -3

Check:

• |2(4) - 1| = |7| = 7 ✓
• |2(-3) - 1| = |-7| = 7 ✓

Solution: x = 4 or x = -3

Example 3: |3x + 5| = 4

Case 1: 3x + 5 = 4 3x = -1 x = -1/3

Case 2: 3x + 5 = -4 3x = -9 x = -3

Solution: x = -1/3 or x = -3

Isolating the Absolute Value First

If the absolute value is not already isolated, isolate it BEFORE setting up two equations.

Example 1: 2|x - 3| = 10

Step 1: Divide both sides by 2 |x - 3| = 5

Step 2: Set up two equations x - 3 = 5 or x - 3 = -5

Step 3: Solve x = 8 or x = -2

Example 2: |x + 1| + 5 = 12

Step 1: Subtract 5 from both sides |x + 1| = 7

Step 2: Set up two equations x + 1 = 7 or x + 1 = -7

Step 3: Solve x = 6 or x = -8

Example 3: 3|2x - 4| - 5 = 10

Step 1: Add 5 to both sides 3|2x - 4| = 15

Step 2: Divide by 3 |2x - 4| = 5

Step 3: Set up two equations 2x - 4 = 5 or 2x - 4 = -5

Step 4: Solve 2x = 9 or 2x = -1 x = 4.5 or x = -0.5

Special Cases

Case 1: |expression| = 0 Only ONE solution (the expression equals 0)

Example: |x - 5| = 0 x - 5 = 0 x = 5 (only solution)

Case 2: |expression| = negative number NO SOLUTION (absolute value cannot be negative)

Example: |x + 2| = -3 No solution (absolute value is never negative)

Case 3: Expression equals its negative Example: |2x - 6| = 6 - 2x This happens when the expression is ≤ 0

2x - 6 = 6 - 2x 4x = 12 x = 3

Check: |2(3) - 6| = |0| = 0 and 6 - 2(3) = 0 ✓

Absolute Value with Variables on Both Sides

Form: |ax + b| = cx + d

This is more complex. Consider when the right side can be positive or negative.

Example: |x - 2| = x + 4

Case 1: x - 2 = x + 4 -2 = 4 (no solution from this case)

Case 2: x - 2 = -(x + 4) x - 2 = -x - 4 2x = -2 x = -1

Check: |-1 - 2| = |-3| = 3 and -1 + 4 = 3 ✓

Solution: x = -1

Solving Equations with Two Absolute Values

Form: |expression₁| = |expression₂|

This means the expressions are equal OR opposites.

Set up two equations:

• expression₁ = expression₂
• expression₁ = -(expression₂)

Example: |2x - 1| = |x + 3|

Case 1: 2x - 1 = x + 3 x = 4

Case 2: 2x - 1 = -(x + 3) 2x - 1 = -x - 3 3x = -2 x = -2/3

Check both solutions in original equation:

• For x = 4: |2(4) - 1| = |7| = 7 and |4 + 3| = |7| = 7 ✓
• For x = -2/3: |2(-2/3) - 1| = |-7/3| = 7/3 and |-2/3 + 3| = |7/3| = 7/3 ✓

Solution: x = 4 or x = -2/3

Graphical Interpretation

The solutions to |x - a| = b are the x-values where the distance from x to a equals b.

Example: |x - 3| = 5 Find all x-values that are 5 units away from 3 x = 3 + 5 = 8 or x = 3 - 5 = -2

On a number line: -2 is 5 units to the left of 3 8 is 5 units to the right of 3

Common Mistakes to Avoid

1. Forgetting the negative case Wrong: |x + 2| = 5, so x + 2 = 5, x = 3 (missing x = -7) Right: Set up BOTH x + 2 = 5 and x + 2 = -5

2. Not isolating absolute value first For 2|x| + 3 = 11, must get |x| = 4 before setting up two equations

3. Thinking absolute value can be negative |x| = -5 has NO solution

4. Not checking solutions Sometimes algebraic solutions don't work in original equation

5. Distributing negative incorrectly -(2x + 3) = -2x - 3, not -2x + 3

Checking Your Solutions

Always substitute solutions back into the ORIGINAL equation.

Example: Solve |2x - 3| = 5 Solutions: x = 4 or x = -1

Check x = 4: |2(4) - 3| = |8 - 3| = |5| = 5 ✓

Check x = -1: |2(-1) - 3| = |-2 - 3| = |-5| = 5 ✓

Both solutions work!

Real-World Applications

Absolute value equations model situations involving distance, tolerance, and error.

Example 1: Manufacturing Tolerance A bolt must be 5 cm long with a tolerance of ±0.2 cm. |length - 5| ≤ 0.2 Acceptable lengths: 4.8 cm to 5.2 cm

Example 2: Temperature Range The temperature should be 70°F, varying by at most 3°. |T - 70| ≤ 3 Acceptable range: 67°F to 73°F

Example 3: Distance Two cars start from the same point and drive in opposite directions. When are they 100 miles apart? If they each travel x miles: |x - (-x)| = |2x| = 100 So 2x = 100, x = 50 miles each

Problem-Solving Strategy

1. Isolate the absolute value expression
2. Check if the right side is positive, zero, or negative
   • Negative: no solution
   • Zero: one solution
   • Positive: usually two solutions
3. Set up two equations (positive and negative cases)
4. Solve both equations
5. Check all solutions in the original equation
6. Reject any solutions that don't work

Absolute Value Inequalities (Preview)

While this topic focuses on equations, you'll later learn:

• |x| < a means -a < x < a (between)
• |x| > a means x < -a or x > a (outside)

Quick Reference

| Equation Type | Setup | Number of Solutions | |---------------|-------|---------------------| | |x| = a (a > 0) | x = ±a | 2 solutions | | |x| = 0 | x = 0 | 1 solution | | |x| = a (a < 0) | No solution | 0 solutions | | |ax + b| = c | ax + b = ±c | Usually 2 | | |expr₁| = |expr₂| | expr₁ = ±expr₂ | Usually 2 |

Practice Tips

• Always isolate the absolute value first
• Set up both positive and negative cases
• Be systematic in your work
• Always check your solutions
• Remember: absolute value is never negative
• Draw number lines to visualize distance
• Look for special cases (0, negative, variables on both sides)
• Practice with different types of problems