Tier 2: Inequalities & Absolute Values

🧠 What Are Inequalities?

Inequalities compare two expressions using symbols like: <, >, ≤, ≥

To solve, treat it like an equation, but remember:

• If you multiply or divide by a negative number, flip the inequality sign!

✏️ Examples

• x + 5 < 10 → Subtract 5 → x < 5
• -2x > 6 → Divide by -2 and flip → x < -3

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🧠 What Are Compound Inequalities?

Compound inequalities contain two inequality statements joined by AND or OR.

✅ AND Example

3 < x + 1 ≤ 7

Step 1: Subtract 1 across all parts → 2 < x ≤ 6

Solution: x is greater than 2 and less than or equal to 6

✅ OR Example

x < -2 OR x > 4

This means x is either less than -2 or greater than 4.

✅ Try These

1. -2 < x - 3 < 4 → Answer: 1 < x < 7
2. x ≤ -1 OR x > 5 → Answer: all values ≤ -1 or > 5

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🧠 What Is Absolute Value?

The absolute value of a number is its distance from 0 on the number line — it is always positive or zero.

📏 Key Property

|a| = a if a ≥ 0; |a| = -a if a < 0

✏️ Examples

• |7| = 7
• |-4| = 4
• |0| = 0

✅ Try These

1. |-8| → Answer: 8
2. |3 - 5| → Answer: 2
3. |2 - 9| → Answer: 7

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🧠 What Is an Absolute Value Equation?

It’s an equation where the unknown is inside an absolute value: |expression| = number.

These always give two solutions — one positive, one negative case.

✏️ Rule:

If |A| = B, then A = B or A = -B

Important: B must be ≥ 0. If B < 0, there is no solution.

✔️ Example 1

|x - 2| = 7

Case 1: x - 2 = 7 → x = 9
Case 2: x - 2 = -7 → x = -5
✅ Solution: x = 9 or x = -5

✔️ Example 2

|3x + 1| = 10

Case 1: 3x + 1 = 10 → x = 3
Case 2: 3x + 1 = -10 → x = -11/3
✅ Solution: x = 3 or x = -11/3

✔️ Example 3

|2x - 5| = -3

Absolute value cannot equal a negative number. ❌ No solution.

✔️ Example 4

|x + 4| = 0

Only one solution: x + 4 = 0 → x = -4

✅ Try These

1. |x - 3| = 8 → Answer: x = 11 or x = -5
2. |2x + 1| = 5 → Answer: x = 2 or x = -3
3. |4x - 2| = -6 → Answer: No solution
4. |x| = 0 → Answer: x = 0

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🧠 Absolute Value Inequalities

These describe a range of values based on distance from zero or a number.

📘 Two Key Rules:

• |A| < B → -B < A < B (AND situation)
• |A| > B → A < -B OR A > B (OR situation)

Note: The value of B must be non-negative. If B < 0, there is no solution for <, and all real numbers for >.

✏️ Example 1

|x - 3| < 5 → -5 < x - 3 < 5 → Add 3 → -2 < x < 8

✏️ Example 2

|2x + 1| ≥ 7 → 2x + 1 ≤ -7 OR 2x + 1 ≥ 7
Solve both → x ≤ -4 OR x ≥ 3

✏️ Example 3

|x| > -2 → Always true, since absolute values are always ≥ 0 → Solution: all real numbers

✏️ Example 4

|x + 2| < 0 → No solution (absolute values cannot be negative)

✅ Try These

1. |x - 4| < 6 → Answer: -2 < x < 10
2. |x + 5| ≥ 3 → Answer: x ≤ -8 OR x ≥ -2
3. |2x - 1| > -3 → Answer: all real numbers
4. |x - 7| < 0 → Answer: no solution

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