⚖Advanced Inequalities

Two Types

• AND (Conjunction): Both conditions must be true
  Example: $x > 2$ AND $x < 7$ means $2 < x < 7$
• OR (Disjunction): At least one condition must be true
  Example: $x < -3$ OR $x> 5$

Real-World Context

🌡️ "The temperature must be between 60°F and 80°F"

This is: $60 \leq T \leq 80$ (an AND compound inequality)

🎢 "You must be under 4 feet tall OR over 6 feet tall for this ride"

This is: $h < 4$ OR $h> 6$ (an OR compound inequality)

Solving AND Inequalities

Method 1: Solve each inequality separately, then find the intersection

Method 2: Work with all parts together if they're written as one statement

Example 1: Solve $-3 < 2x + 1 \leq 7$

Step 1: Subtract 1 from all parts

$-4 < 2x \leq 6$

Step 2: Divide all parts by 2

$$-2 < x \leq 3$$

Interpretation: $x$ is greater than -2 but less than or equal to 3

📊 On a number line: open circle at -2, closed circle at 3, shade between

Example 2: Solve $x + 5 > 3$ AND $x - 2 < 4$

Solve separately:

• $x + 5 > 3$ → $x > -2$
• $x - 2 < 4$ → $x < 6$

$$-2 < x < 6$$

Example 1: Solve $x < -1$ OR $x> 3$

The solution has TWO separate parts:

• All numbers less than -1
• All numbers greater than 3

Solution: $(-\infty, -1) \cup (3, \infty)$

📊 On a number line: shade left of -1 AND shade right of 3

Example 2: Solve $2x + 1 \leq -3$ OR $3x - 2 > 7$

Part 1: $2x + 1 \leq -3$

$2x \leq -4$ → $x \leq -2$

Part 2: $3x - 2 > 7$

$3x > 9$ → $x > 3$

Final Solution: $x \leq -2$ OR $x > 3$ ✨

Key Patterns

Pattern 1: $|x| < a$ (distance less than $a$)

Rewrite as: $-a < x < a$ (AND)

Pattern 2: $|x| > a$ (distance greater than $a$)

Rewrite as: $x < -a$ OR $x> a$

Example 1: Solve $|2x - 5| < 3$

Rewrite: $-3 < 2x - 5 < 3$

Add 5: $2 < 2x < 8$

Divide by 2: $1 < x < 4$

Example 2: Solve $|x + 2| \geq 5$

Rewrite: $x + 2 \leq -5$ OR $x + 2 \geq 5$

Solve: $x \leq -7$ OR $x \geq 3$ ✨

Example 3: Solve $|3x - 1| \leq 8$

$-8 \leq 3x - 1 \leq 8$

$-7 \leq 3x \leq 9$

$$-\frac{7}{3} \leq x \leq 3$$

Graphing Symbols

• Open circle ○: Use for $<$ or $>$ (endpoint NOT included)
• Closed circle ●: Use for $\leq$ or $\geq$ (endpoint IS included)
• Solid line: Shows all numbers in the solution set
• Arrow: Indicates the solution continues forever in that direction

Example: Graph $-2 \leq x < 5$

Draw a number line with:

• Closed circle at -2 (included)
• Open circle at 5 (not included)
• Solid line connecting them

Example: Graph $x < -3$ OR $x \geq 1$

Two separate regions:

• Arrow pointing left from open circle at -3
• Arrow pointing right from closed circle at 1

Case 1: No Solution

$x > 5$ AND $x < 2$ has NO solution!

(Nothing can be both greater than 5 AND less than 2)

Case 2: All Real Numbers

$x > 5$ OR $x < 10$ is ALL real numbers!

(Every number satisfies at least one condition)

Case 3: Simplification

$x < 7$ AND $x < 3$ simplifies to just $x < 3$

(The more restrictive condition)

Case 4: Absolute Value Edge Cases

$|x| < -5$ has NO solution (distance can't be negative!)

$|x| > -2$ is ALL real numbers (distance is always ≥ 0)