Absolute Value Equations and Inequalities

Review: Absolute Value

Absolute value: Distance from zero on number line

Notation: |x|

Always non-negative: |x| ≥ 0

Examples:

• |5| = 5
• |-5| = 5
• |0| = 0

Distance Interpretation

|x| = 3 means "x is 3 units from 0"

Two solutions: x = 3 or x = -3

Visual: Both 3 and -3 are distance 3 from 0

Solving Simple Absolute Value Equations

If |x| = a (where a ≥ 0), then x = a or x = -a

Example 1: Basic Equation

Solve: |x| = 7

Two solutions:

x = 7  or  x = -7

Answer: x = 7 or x = -7

Example 2: No Solution

Solve: |x| = -4

Absolute value cannot be negative!

Answer: No solution

Example 3: Zero

Solve: |x| = 0

Only distance 0 from zero is 0 itself

Answer: x = 0

Absolute Value Equations with Expressions

If |ax + b| = c, then ax + b = c or ax + b = -c

Example 1: One-Step Inside

Solve: |x + 3| = 5

Set up two equations:

x + 3 = 5  or  x + 3 = -5

Solve each:

x = 2  or  x = -8

Check:

• |2 + 3| = |5| = 5 ✓
• |-8 + 3| = |-5| = 5 ✓

Answer: x = 2 or x = -8

Example 2: Coefficient Inside

Solve: |2x - 4| = 10

Two equations:

2x - 4 = 10  or  2x - 4 = -10

Solve:

2x = 14       2x = -6
x = 7         x = -3

Answer: x = 7 or x = -3

Example 3: Isolate First

Solve: |x - 1| + 3 = 8

Isolate absolute value:

|x - 1| = 5

Two equations:

x - 1 = 5  or  x - 1 = -5
x = 6          x = -4

Answer: x = 6 or x = -4

Equations with No Solution

When absolute value equals negative number

Example: No Solution Case

Solve: |x + 2| = -3

Absolute value is never negative

Answer: No solution

Example: Isolate Reveals Impossibility

Solve: |x| + 5 = 2

Isolate:

|x| = -3

Cannot be negative

Answer: No solution

Absolute Value Inequalities: Less Than

|x| < a means "distance from 0 is less than a"

Solution: -a < x < a (between -a and a)

Example 1: Basic Inequality

Solve: |x| < 4

Distance from 0 less than 4:

-4 < x < 4

Number line: Open circles at -4 and 4, shade between

Answer: -4 < x < 4

Example 2: With Expression

Solve: |x + 2| < 5

Compound inequality:

-5 < x + 2 < 5

Subtract 2 from all parts:

-7 < x < 3

Answer: -7 < x < 3

Example 3: Coefficient

Solve: |2x - 3| ≤ 7

Set up compound:

-7 ≤ 2x - 3 ≤ 7

Add 3:

-4 ≤ 2x ≤ 10

Divide by 2:

-2 ≤ x ≤ 5

Answer: -2 ≤ x ≤ 5

Absolute Value Inequalities: Greater Than

|x| > a means "distance from 0 is greater than a"

Solution: x < -a or x > a (outside the interval)

Example 1: Basic Greater Than

Solve: |x| > 3

Distance more than 3 from 0:

x < -3  or  x > 3

Number line: Arrows going left from -3 and right from 3

Answer: x < -3 or x > 3

Example 2: With Expression

Solve: |x - 4| > 6

Two separate inequalities:

x - 4 < -6  or  x - 4 > 6

Solve each:

x < -2  or  x > 10

Answer: x < -2 or x > 10

Example 3: Isolate First

Solve: |x + 1| - 2 ≥ 3

Isolate:

|x + 1| ≥ 5

Split:

x + 1 ≤ -5  or  x + 1 ≥ 5
x ≤ -6      or  x ≥ 4

Answer: x ≤ -6 or x ≥ 4

Summary of Inequality Rules

|x| < a: -a < x < a (AND compound)

|x| > a: x < -a or x > a (OR statement)

|x| ≤ a: -a ≤ x ≤ a (AND compound, inclusive)

|x| ≥ a: x ≤ -a or x ≥ a (OR statement, inclusive)

Graphing Solutions

Less than: One continuous interval (between)

Greater than: Two rays (outside)

Example: Graph |x - 1| ≤ 3

Solve:

-3 ≤ x - 1 ≤ 3
-2 ≤ x ≤ 4

Graph: Closed circles at -2 and 4, shade between

Special Cases

Absolute value always ≥ 0

Example 1: Always True

Solve: |x| > -2

Since |x| ≥ 0, always greater than -2

Answer: All real numbers

Example 2: Sometimes Impossible

Solve: |x| < -1

Absolute value never negative

Answer: No solution

Absolute Value Equations with Two Absolute Values

|a| = |b| means a = b or a = -b

Example: Two Absolute Values

Solve: |x + 2| = |x - 4|

Case 1: x + 2 = x - 4

2 = -4  (false, no solution from this case)

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

x + 2 = -x + 4
2x = 2
x = 1

Answer: x = 1

Real-World Applications

Tolerance: Manufacturing specifications

• Part size: |x - 5| ≤ 0.1 cm

Error margin: Measurements

• Actual vs. expected: |actual - expected| < error

Temperature range:

• |T - 72| ≤ 3 means 69°F to 75°F

Example: Quality Control

Bolt length must be 8 cm ± 0.2 cm. Write inequality.

Distance from 8 no more than 0.2:

|L - 8| ≤ 0.2

Solution:

-0.2 ≤ L - 8 ≤ 0.2
7.8 ≤ L ≤ 8.2

Acceptable range: 7.8 cm to 8.2 cm

Strategy for Solving

Equations:

1. Isolate absolute value
2. Check if right side is non-negative
3. Split into two equations
4. Solve both
5. Check solutions

Inequalities:

1. Isolate absolute value
2. Determine if "less than" (AND) or "greater than" (OR)
3. Set up compound inequality or two separate inequalities
4. Solve
5. Graph if needed

Practice