Absolute Value Equations and Inequalities - Ninth Grade Math
Introduction to Absolute Value
Absolute Value: The distance of a number from zero on a number line
Symbol: $|x|$ (read as "the absolute value of x")
Key Property: Always non-negative (≥ 0)
Examples: $|5| = 5$, $|-5| = 5$, $|0| = 0$
Symbol: $|x|$ (read as "the absolute value of x")
Key Property: Always non-negative (≥ 0)
Examples: $|5| = 5$, $|-5| = 5$, $|0| = 0$
Absolute Value Definition:
$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
In words:
• If the number is positive or zero, absolute value is the number itself
• If the number is negative, absolute value is its opposite (positive)
$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
In words:
• If the number is positive or zero, absolute value is the number itself
• If the number is negative, absolute value is its opposite (positive)
1. Solve Absolute Value Equations
Absolute Value Equation: An equation containing an absolute value expression
General Form: $|x| = a$ or $|ax + b| = c$
Key Concept: If $|x| = a$, then $x = a$ OR $x = -a$ (two solutions)
General Form: $|x| = a$ or $|ax + b| = c$
Key Concept: If $|x| = a$, then $x = a$ OR $x = -a$ (two solutions)
Basic Absolute Value Property:
If $|X| = a$ where $a > 0$, then:
$$X = a \quad \text{OR} \quad X = -a$$
Special Cases:
• If $|X| = 0$, then $X = 0$ (one solution)
• If $|X| = a$ where $a < 0$, then NO SOLUTION (absolute value can't be negative)
If $|X| = a$ where $a > 0$, then:
$$X = a \quad \text{OR} \quad X = -a$$
Special Cases:
• If $|X| = 0$, then $X = 0$ (one solution)
• If $|X| = a$ where $a < 0$, then NO SOLUTION (absolute value can't be negative)
Steps to Solve Absolute Value Equations:
Step 1: Isolate the absolute value expression on one side
Step 2: Check the number on the other side:
• If negative → No solution
• If zero → Remove absolute value bars and solve
• If positive → Create two equations
Step 3: Solve both equations separately
Step 4: Check both solutions in original equation
Step 1: Isolate the absolute value expression on one side
Step 2: Check the number on the other side:
• If negative → No solution
• If zero → Remove absolute value bars and solve
• If positive → Create two equations
Step 3: Solve both equations separately
Step 4: Check both solutions in original equation
Example 1: Solve $|x| = 7$
Already isolated, positive number on right
Two equations:
$x = 7$ OR $x = -7$
Check: $|7| = 7$ ✓ and $|-7| = 7$ ✓
Answer: $x = 7$ or $x = -7$
Already isolated, positive number on right
Two equations:
$x = 7$ OR $x = -7$
Check: $|7| = 7$ ✓ and $|-7| = 7$ ✓
Answer: $x = 7$ or $x = -7$
Example 2: Solve $|x - 3| = 5$
Already isolated
Two equations:
$x - 3 = 5$ OR $x - 3 = -5$
$x = 8$ OR $x = -2$
Check: $|8-3| = |5| = 5$ ✓ and $|-2-3| = |-5| = 5$ ✓
Answer: $x = 8$ or $x = -2$
Already isolated
Two equations:
$x - 3 = 5$ OR $x - 3 = -5$
$x = 8$ OR $x = -2$
Check: $|8-3| = |5| = 5$ ✓ and $|-2-3| = |-5| = 5$ ✓
Answer: $x = 8$ or $x = -2$
Example 3: Solve $|2x + 1| = 9$
Two equations:
$2x + 1 = 9$ OR $2x + 1 = -9$
$2x = 8$ OR $2x = -10$
$x = 4$ OR $x = -5$
Check: $|2(4)+1| = |9| = 9$ ✓ and $|2(-5)+1| = |-9| = 9$ ✓
Answer: $x = 4$ or $x = -5$
Two equations:
$2x + 1 = 9$ OR $2x + 1 = -9$
$2x = 8$ OR $2x = -10$
$x = 4$ OR $x = -5$
Check: $|2(4)+1| = |9| = 9$ ✓ and $|2(-5)+1| = |-9| = 9$ ✓
Answer: $x = 4$ or $x = -5$
Example 4: Solve $|x - 4| + 3 = 10$
Step 1: Isolate absolute value
$|x - 4| = 7$
Step 2: Two equations
$x - 4 = 7$ OR $x - 4 = -7$
$x = 11$ OR $x = -3$
Answer: $x = 11$ or $x = -3$
Step 1: Isolate absolute value
$|x - 4| = 7$
Step 2: Two equations
$x - 4 = 7$ OR $x - 4 = -7$
$x = 11$ OR $x = -3$
Answer: $x = 11$ or $x = -3$
Example 5: Solve $2|x + 5| - 6 = 8$
Isolate:
$2|x + 5| = 14$
$|x + 5| = 7$
Two equations:
$x + 5 = 7$ OR $x + 5 = -7$
$x = 2$ OR $x = -12$
Answer: $x = 2$ or $x = -12$
Isolate:
$2|x + 5| = 14$
$|x + 5| = 7$
Two equations:
$x + 5 = 7$ OR $x + 5 = -7$
$x = 2$ OR $x = -12$
Answer: $x = 2$ or $x = -12$
Example 6: Solve $|3x - 2| + 5 = 3$
Isolate:
$|3x - 2| = -2$
Absolute value cannot equal negative number!
Answer: No solution
Isolate:
$|3x - 2| = -2$
Absolute value cannot equal negative number!
Answer: No solution
Example 7: Solve $|2x - 8| = 0$
When absolute value = 0:
$2x - 8 = 0$
$2x = 8$
$x = 4$
Answer: $x = 4$ (one solution)
When absolute value = 0:
$2x - 8 = 0$
$2x = 8$
$x = 4$
Answer: $x = 4$ (one solution)
2. Graph Solutions to Absolute Value Equations
Graphing on Number Line: Showing solutions as points on a number line
Notation: Use solid dots for exact values
Purpose: Visual representation of solutions
Notation: Use solid dots for exact values
Purpose: Visual representation of solutions
Graphing Rules:
• Use solid/filled dots for solutions (exact values)
• Label each solution point with its value
• If no solution: write "No Solution" or leave number line blank
• If one solution: mark one point
• If two solutions: mark two points
• Use solid/filled dots for solutions (exact values)
• Label each solution point with its value
• If no solution: write "No Solution" or leave number line blank
• If one solution: mark one point
• If two solutions: mark two points
Example 1: Graph solutions to $|x| = 4$
Solutions: $x = 4$ and $x = -4$
Number Line Graph:
``` ←─────●─────────────●─────→ -4 4 ```
Two solid dots at -4 and 4
Solutions: $x = 4$ and $x = -4$
Number Line Graph:
``` ←─────●─────────────●─────→ -4 4 ```
Two solid dots at -4 and 4
Example 2: Graph solutions to $|x - 2| = 3$
Solve:
$x - 2 = 3$ → $x = 5$
$x - 2 = -3$ → $x = -1$
Number Line Graph:
``` ←─────●─────────────●─────→ -1 5 ```
Two solid dots at -1 and 5
Solve:
$x - 2 = 3$ → $x = 5$
$x - 2 = -3$ → $x = -1$
Number Line Graph:
``` ←─────●─────────────●─────→ -1 5 ```
Two solid dots at -1 and 5
Example 3: Graph solutions to $|x + 1| = 0$
Solve: $x + 1 = 0$ → $x = -1$
Number Line Graph:
``` ←─────────●───────────────→ -1 ```
One solid dot at -1
Solve: $x + 1 = 0$ → $x = -1$
Number Line Graph:
``` ←─────────●───────────────→ -1 ```
One solid dot at -1
3. Write Absolute Value Equations from Graphs
Reverse Process: Given points on number line, write the equation
Key Insight: Solutions are equidistant from a center point
Center Point: Midpoint between two solutions
Key Insight: Solutions are equidistant from a center point
Center Point: Midpoint between two solutions
Formula for Writing Equations:
Given two solutions $a$ and $b$ (where $a < b$):
Step 1: Find the center
$$\text{Center} = \frac{a + b}{2}$$
Step 2: Find the distance from center
$$\text{Distance} = \frac{b - a}{2}$$
Step 3: Write equation
$$|x - \text{center}| = \text{distance}$$
Given two solutions $a$ and $b$ (where $a < b$):
Step 1: Find the center
$$\text{Center} = \frac{a + b}{2}$$
Step 2: Find the distance from center
$$\text{Distance} = \frac{b - a}{2}$$
Step 3: Write equation
$$|x - \text{center}| = \text{distance}$$
Steps to Write Equation from Graph:
Step 1: Identify the two solution points
Step 2: Find midpoint (center) of the two points
Step 3: Find distance from center to either point
Step 4: Write as $|x - \text{center}| = \text{distance}$
Step 1: Identify the two solution points
Step 2: Find midpoint (center) of the two points
Step 3: Find distance from center to either point
Step 4: Write as $|x - \text{center}| = \text{distance}$
Example 1: Write equation for points at -3 and 5
Center: $\frac{-3 + 5}{2} = \frac{2}{2} = 1$
Distance: $\frac{5 - (-3)}{2} = \frac{8}{2} = 4$
(or distance from 1 to 5 is 4)
Equation: $|x - 1| = 4$
Center: $\frac{-3 + 5}{2} = \frac{2}{2} = 1$
Distance: $\frac{5 - (-3)}{2} = \frac{8}{2} = 4$
(or distance from 1 to 5 is 4)
Equation: $|x - 1| = 4$
Example 2: Write equation for points at -6 and 2
Center: $\frac{-6 + 2}{2} = \frac{-4}{2} = -2$
Distance: $\frac{2 - (-6)}{2} = \frac{8}{2} = 4$
Equation: $|x - (-2)| = 4$ or $|x + 2| = 4$
Center: $\frac{-6 + 2}{2} = \frac{-4}{2} = -2$
Distance: $\frac{2 - (-6)}{2} = \frac{8}{2} = 4$
Equation: $|x - (-2)| = 4$ or $|x + 2| = 4$
Example 3: Write equation for points at 1 and 7
Center: $\frac{1 + 7}{2} = \frac{8}{2} = 4$
Distance: $\frac{7 - 1}{2} = \frac{6}{2} = 3$
Equation: $|x - 4| = 3$
Center: $\frac{1 + 7}{2} = \frac{8}{2} = 4$
Distance: $\frac{7 - 1}{2} = \frac{6}{2} = 3$
Equation: $|x - 4| = 3$
Example 4: Write equation for single point at 5
Only one solution means distance = 0
Equation: $|x - 5| = 0$
Only one solution means distance = 0
Equation: $|x - 5| = 0$
4. Solve Absolute Value Inequalities
Absolute Value Inequality: Inequality with absolute value expression
Types: $|x| < a$, $|x| \leq a$, $|x| > a$, $|x| \geq a$
Key Difference: Solutions are ranges, not just points
Types: $|x| < a$, $|x| \leq a$, $|x| > a$, $|x| \geq a$
Key Difference: Solutions are ranges, not just points
Absolute Value Inequality Rules:
LESS THAN (< or ≤) - "AND" Statement:
If $|X| < a$ where $a > 0$, then:
$$-a < X < a$$
Solution is BETWEEN -a and a
If $|X| \leq a$ where $a > 0$, then:
$$-a \leq X \leq a$$
GREATER THAN (> or ≥) - "OR" Statement:
If $|X| > a$ where $a > 0$, then:
$$X < -a \quad \text{OR} \quad X > a$$
Solution is OUTSIDE -a and a
If $|X| \geq a$ where $a > 0$, then:
$$X \leq -a \quad \text{OR} \quad X \geq a$$
LESS THAN (< or ≤) - "AND" Statement:
If $|X| < a$ where $a > 0$, then:
$$-a < X < a$$
Solution is BETWEEN -a and a
If $|X| \leq a$ where $a > 0$, then:
$$-a \leq X \leq a$$
GREATER THAN (> or ≥) - "OR" Statement:
If $|X| > a$ where $a > 0$, then:
$$X < -a \quad \text{OR} \quad X > a$$
Solution is OUTSIDE -a and a
If $|X| \geq a$ where $a > 0$, then:
$$X \leq -a \quad \text{OR} \quad X \geq a$$
Memory Aid:
• LESS THAN: Solution is in the MIDDLE (between) → AND
• GREATER THAN: Solution is on the SIDES (outside) → OR
• LESS THAN: Solution is in the MIDDLE (between) → AND
• GREATER THAN: Solution is on the SIDES (outside) → OR
Solving "Less Than" Inequalities
Example 1: Solve $|x| < 5$
Apply rule: $-5 < x < 5$
Interval notation: $(-5, 5)$
Answer: All numbers between -5 and 5
Apply rule: $-5 < x < 5$
Interval notation: $(-5, 5)$
Answer: All numbers between -5 and 5
Example 2: Solve $|x - 3| \leq 4$
Apply rule:
$-4 \leq x - 3 \leq 4$
Add 3 to all parts:
$-4 + 3 \leq x \leq 4 + 3$
$-1 \leq x \leq 7$
Interval notation: $[-1, 7]$
Answer: All numbers from -1 to 7, inclusive
Apply rule:
$-4 \leq x - 3 \leq 4$
Add 3 to all parts:
$-4 + 3 \leq x \leq 4 + 3$
$-1 \leq x \leq 7$
Interval notation: $[-1, 7]$
Answer: All numbers from -1 to 7, inclusive
Example 3: Solve $|2x + 1| < 7$
Apply rule:
$-7 < 2x + 1 < 7$
Subtract 1:
$-8 < 2x < 6$
Divide by 2:
$-4 < x < 3$
Answer: $(-4, 3)$
Apply rule:
$-7 < 2x + 1 < 7$
Subtract 1:
$-8 < 2x < 6$
Divide by 2:
$-4 < x < 3$
Answer: $(-4, 3)$
Solving "Greater Than" Inequalities
Example 4: Solve $|x| > 3$
Apply rule:
$x < -3$ OR $x > 3$
Interval notation: $(-\infty, -3) \cup (3, \infty)$
Answer: Numbers less than -3 OR greater than 3
Apply rule:
$x < -3$ OR $x > 3$
Interval notation: $(-\infty, -3) \cup (3, \infty)$
Answer: Numbers less than -3 OR greater than 3
Example 5: Solve $|x + 2| \geq 5$
Apply rule:
$x + 2 \leq -5$ OR $x + 2 \geq 5$
Solve each:
$x \leq -7$ OR $x \geq 3$
Interval notation: $(-\infty, -7] \cup ←─────○═══════════○─────→ -4 4 ``` Open circles at -4 and 4, shaded between
Apply rule:
$x + 2 \leq -5$ OR $x + 2 \geq 5$
Solve each:
$x \leq -7$ OR $x \geq 3$
Interval notation: $(-\infty, -7] \cup ←─────○═══════════○─────→ -4 4 ``` Open circles at -4 and 4, shaded between
Example 2: Graph $|x + 1| \leq 3$
Solution: $-4 \leq x \leq 2$
Number Line:
``` ←─────●═══════════●─────→ -4 2 ``` Closed circles at -4 and 2, shaded between
Solution: $-4 \leq x \leq 2$
Number Line:
``` ←─────●═══════════●─────→ -4 2 ``` Closed circles at -4 and 2, shaded between
Example 3: Graph $|x| > 2$
Solution: $x < -2$ OR $x > 2$
Number Line:
``` ←═════○─────────────○═════→ -2 2 ``` Open circles at -2 and 2, shaded on both outside regions
Solution: $x < -2$ OR $x > 2$
Number Line:
``` ←═════○─────────────○═════→ -2 2 ``` Open circles at -2 and 2, shaded on both outside regions
Example 4: Graph $|x - 1| \geq 3$
Solve:
$x - 1 \leq -3$ OR $x - 1 \geq 3$
$x \leq -2$ OR $x \geq 4$
Number Line:
``` ←═════●─────────────●═════→ -2 4 ``` Closed circles at -2 and 4, shaded on both outside regions
Solve:
$x - 1 \leq -3$ OR $x - 1 \geq 3$
$x \leq -2$ OR $x \geq 4$
Number Line:
``` ←═════●─────────────●═════→ -2 4 ``` Closed circles at -2 and 4, shaded on both outside regions
Example 5: Graph $|2x - 4| < 6$
Solve:
$-6 < 2x - 4 < 6$
$-2 < 2x < 10$
$-1 < x < 5$
Number Line:
``` ←─────○═══════════○─────→ -1 5 ``` Open circles at -1 and 5, shaded between
Solve:
$-6 < 2x - 4 < 6$
$-2 < 2x < 10$
$-1 < x < 5$
Number Line:
``` ←─────○═══════════○─────→ -1 5 ``` Open circles at -1 and 5, shaded between
Absolute Value Equations Summary
| Equation Type | Solution Method | Number of Solutions |
|---|---|---|
| $|X| = a$ where $a > 0$ | $X = a$ OR $X = -a$ | Two solutions |
| $|X| = 0$ | $X = 0$ | One solution |
| $|X| = a$ where $a < 0$ | No solution (impossible) | Zero solutions |
Absolute Value Inequalities Summary
| Inequality | Solution Form | Graph Type | Word Description |
|---|---|---|---|
| $|X| < a$ | $-a < X < a$ | Shade between (○─○) | Between -a and a |
| $|X| \leq a$ | $-a \leq X \leq a$ | Shade between (●─●) | From -a to a, inclusive |
| $|X| > a$ | $X < -a$ OR $X > a$ | Shade outside (○ ○) | Less than -a OR greater than a |
| $|X| \geq a$ | $X \leq -a$ OR $X \geq a$ | Shade outside (● ●) | At most -a OR at least a |
Graphing Symbols Reference
| Symbol | Meaning | Use With | Visual |
|---|---|---|---|
| ○ Open Circle | Endpoint NOT included | < or > | Hollow/empty circle |
| ● Closed Circle | Endpoint IS included | ≤ or ≥ | Filled/solid circle |
| ═══ Shading | All values in range | Any inequality | Thick line or color |
| → Arrow | Continues forever | Unbounded intervals | Points to infinity |
Step-by-Step Decision Tree
| Question | If YES | If NO |
|---|---|---|
| Is it an equation (=)? | Use equation rules: $X = a$ OR $X = -a$ | Go to next question |
| Is it less than (< or ≤)? | Solution BETWEEN: $-a < X < a$ | Go to next question |
| Is it greater than (> or ≥)? | Solution OUTSIDE: $X < -a$ OR $X > a$ | Check problem |
| Is right side negative? | No solution (equations) or All real numbers (>) or No solution (<) | Proceed normally |
Quick Reference: Interval Notation
| Inequality | Interval Notation | Description |
|---|---|---|
| $a < x < b$ | $(a, b)$ | Between a and b, not including endpoints |
| $a \leq x \leq b$ | $[a, b]$ | From a to b, including endpoints |
| $x < a$ OR $x > b$ | $(-\infty, a) \cup (b, \infty)$ | Less than a OR greater than b |
| $x \leq a$ OR $x \geq b$ | $(-\infty, a] \cupThis complete HTML code covers all 5 topics for absolute value equations and inequalities in ninth-grade algebra and is ready to paste directly into your WordPress website! It includes all formulas properly rendered with MathJax, organized with light colors and bold fonts, comprehensive examples, decision trees, and is a complete, self-contained section perfect for student reference. Shares: |