General Form:

\( |ax + b| = c \)

Steps to Solve Absolute Value Equations:

1. Step 1: Isolate the absolute value expression
   • Get \( |ax + b| \) by itself on one side
2. Step 2: Check the value on the other side
   • If c < 0: No solution (absolute value cannot be negative)
   • If c = 0: One solution (set expression = 0)
   • If c > 0: Two solutions (continue to Step 3)
3. Step 3: Write two equations (positive and negative cases)
   • \( ax + b = c \)   AND   \( ax + b = -c \)
4. Step 4: Solve both equations
   • Find x for each equation
   • Check both solutions in the original equation

Solution Formula:

If \( |X| = p \) where \( p > 0 \), then \( X = p \)   OR   \( X = -p \)

Example 1: Simple Equation

Solve: \( |x - 3| = 7 \)

Step 1: Already isolated

Step 2: c = 7 > 0, so two solutions exist

Step 3: Write two equations:

• \( x - 3 = 7 \)

• \( x - 3 = -7 \)

Step 4: Solve both:

• \( x = 10 \)

• \( x = -4 \)

Solution: \( x = 10 \) or \( x = -4 \)

Example 2: Equation Requiring Isolation

Solve: \( 2|3x + 1| - 5 = 9 \)

Step 1: Isolate absolute value:

\( 2|3x + 1| = 14 \)

\( |3x + 1| = 7 \)

Step 3: Write two equations:

• \( 3x + 1 = 7 \) → \( 3x = 6 \) → \( x = 2 \)

• \( 3x + 1 = -7 \) → \( 3x = -8 \) → \( x = -\frac{8}{3} \)

Solution: \( x = 2 \) or \( x = -\frac{8}{3} \)

Example 3: No Solution

Solve: \( |2x - 5| + 3 = 1 \)

Step 1: Isolate: \( |2x - 5| = -2 \)

Step 2: Since -2 < 0, absolute value cannot equal a negative number

Solution: No solution (∅)

Graphing on a Number Line:

Solutions to absolute value equations are represented as points on the number line.

• Solid dots (•) represent the solutions
• For \( |x| = c \), plot points at \( x = c \) and \( x = -c \)
• No solution: No points on number line

Steps to Graph Solutions:

1. Solve the absolute value equation
2. Draw a number line with appropriate scale
3. Mark each solution with a solid dot
4. Label the points with their values

Example: Graph \( |x - 2| = 5 \)

Solve:

• \( x - 2 = 5 \) → \( x = 7 \)

• \( x - 2 = -5 \) → \( x = -3 \)

Graph: Place solid dots at x = -3 and x = 7 on the number line

←———•—————————•———→
    -3             7

Steps to Write Equations from Graphs:

1. Step 1: Identify the solutions from the graph
   • Find the marked points on the number line
   • Let these be \( x_1 \) and \( x_2 \)
2. Step 2: Find the midpoint (center)
   • Midpoint \( h = \frac{x_1 + x_2}{2} \)
3. Step 3: Find the distance from center
   • Distance \( d = \frac{|x_2 - x_1|}{2} \) or \( d = |x_1 - h| \)
4. Step 4: Write the equation
   • \( |x - h| = d \)

Quick Formula:

If solutions are \( x_1 \) and \( x_2 \), then:   \( |x - \frac{x_1+x_2}{2}| = \frac{|x_2-x_1|}{2} \)

Example 1: Write Equation from Graph

Given: Points at x = 2 and x = 8

Step 1: \( x_1 = 2 \), \( x_2 = 8 \)

Step 2: Midpoint: \( h = \frac{2 + 8}{2} = 5 \)

Step 3: Distance: \( d = \frac{|8 - 2|}{2} = 3 \)

Equation: \( |x - 5| = 3 \)

Example 2: Negative Solutions

Given: Points at x = -6 and x = 4

Step 2: Midpoint: \( h = \frac{-6 + 4}{2} = -1 \)

Step 3: Distance: \( d = \frac{|4 - (-6)|}{2} = 5 \)

Equation: \( |x + 1| = 5 \)   (same as \( |x - (-1)| = 5 \))

Two Types of Absolute Value Inequalities:

Key Formulas:

Steps to Solve Absolute Value Inequalities:

1. Step 1: Isolate the absolute value
2. Step 2: Identify the type
   • Less than (< or ≤) → AND compound inequality
   • Greater than (> or ≥) → OR compound inequality
3. Step 3: Rewrite without absolute value
   • For |X| < c: write \( -c < X < c \)
   • For |X| > c: write \( X < -c \) OR \( X > c \)
4. Step 4: Solve and write solution
   • Solve the resulting inequality/inequalities
   • Express in interval notation

Example 1: Less Than (AND)

Solve: \( |x - 3| < 5 \)

Step 1: Already isolated

Step 2: Less than type → AND

Step 3: Rewrite: \( -5 < x - 3 < 5 \)

Step 4: Add 3 to all parts: \( -2 < x < 8 \)

Solution: \( (-2, 8) \) or \( -2 < x < 8 \)

Example 2: Greater Than (OR)

Solve: \( |2x + 1| \geq 7 \)

Step 2: Greater than or equal type → OR

Step 3: Rewrite: \( 2x + 1 \leq -7 \) OR \( 2x + 1 \geq 7 \)

Step 4: Solve each:

• \( 2x \leq -8 \) → \( x \leq -4 \)

• \( 2x \geq 6 \) → \( x \geq 3 \)

Solution: \( (-\infty, -4] \cup [3, \infty) \) or \( x \leq -4 \) or \( x \geq 3 \)

Example 3: With Isolation

Solve: \( 3|x - 2| + 5 > 14 \)

Step 1: Isolate: \( 3|x - 2| > 9 \) → \( |x - 2| > 3 \)

Step 3: \( x - 2 < -3 \) OR \( x - 2 > 3 \)

Step 4: \( x < -1 \) OR \( x > 5 \)

Solution: \( (-\infty, -1) \cup (5, \infty) \)

Graphing on a Number Line:

Solutions to inequalities are shown as intervals (shaded regions) on the number line.

Steps to Graph:

1. Solve the absolute value inequality
2. Draw a number line with appropriate scale
3. Mark boundary points with circles (open or closed)
4. Shade the solution region(s)
5. For OR: shade both regions; For AND: shade middle region

Example 1: Less Than (AND) - Single Interval

Graph: \( |x + 1| \leq 4 \) which gives \( -5 \leq x \leq 3 \)

Solution: \( [-5, 3] \)

←———•═══════•———→
    -5           3
(Closed circles, shade between)

Example 2: Greater Than (OR) - Two Intervals

Graph: \( |x - 2| > 3 \) which gives \( x < -1 \) or \( x > 5 \)

Solution: \( (-\infty, -1) \cup (5, \infty) \)

←═══○———————○═══→
    -1           5
(Open circles, shade both ends)

Graphing Tips:

• AND inequalities: Shade the region BETWEEN the two values (one continuous segment)
• OR inequalities: Shade BOTH regions on the outside (two separate segments)
• Remember: < or > use open circles (○); ≤ or ≥ use closed circles (•)

Key Concepts:

• Absolute Value Equations: If \( |X| = c \) (c > 0), then \( X = c \) OR \( X = -c \)
• Graph Equations: Mark solutions as solid dots on number line
• Write from Graph: Find midpoint h and distance d, write \( |x - h| = d \)
• Less Than Inequalities: \( |X| < c \) → \( -c < X < c \) (AND, one interval)
• Greater Than Inequalities: \( |X| > c \) → \( X < -c \) OR \( X > c \) (OR, two intervals)
• Graph Inequalities: Shade intervals with open (○) or closed (•) circles