1. Basic Idea

An inequality compares two expressions using symbols:

• < → less than

• > → greater than

• ≤ → less than or equal to

• ≥ → greater than or equal to

Example:
$3x+2>11$

🔹 2. Solving Linear Inequalities

Steps:

1. Simplify both sides (remove parentheses, combine like terms).

2. Isolate the variable on one side.

3. Important:
   🔸 When you multiply or divide by a negative number, flip the inequality sign.

Example:

$$-2x+5>9$$

Subtract 5:
$-2x>4$
Divide by -2 (flip sign):

$$x<-2$$

✅ Final Answer: $x<-2$

🔹 3. Graphing on a Number Line

Example:
$x\ge -2$ → Closed circle at -2, shade right

🔹 4. Interval Notation

🔹 5. Compound Inequalities

Two inequalities combined using “AND” or “OR”.

A. AND Inequalities (Intersection)

The solution satisfies both inequalities.
Think: “Between two values.”

Example:

$$3<x\le 7$$

means $x$ is greater than 3 and less than or equal to 7.

Interval Notation: $(3,7]$

Graph: Shade between 3 and 7.

B. OR Inequalities (Union)

The solution satisfies either inequality.
Think: “Outside values.”

Example:

$$x<2\text{ or }x\ge 5$$

means $x$ is less than 2 or greater than or equal to 5.

Interval Notation: $(-\infty ,2)\cup [5,\infty )$

Graph: Shade both sides, not between.

🔹 6. Checking Solutions

• Always test a point from each region (especially for OR inequalities).

• Plug into the original inequality to confirm.

🔹 7. Absolute Value Inequalities

Example:

$$|x-4|<3$$

→ $-3<x-4<3$
Add 4 → $1<x<7$

✅ Answer: $(1,7)$

🔹 8. Special Cases

• $|x|<-a$ → No solution (absolute value can’t be negative).

• $|x|>-a$ → All real numbers (always true).

🔹 9. Compound Inequalities with Fractions

Always clear fractions first!
Example:

2x−34≤5\frac{2x – 3}{4} ≤ 542x−3​≤5

Multiply both sides by 4:
$2x-3\le 20$
→ $x\le 11.5$

🔹 10. Graphing Tricks

• “AND” → Between two points

• “OR” → Outside two points

• Check boundary points carefully (open/closed).

🔹 11. Common Mistakes to Avoid

❌ Forgetting to flip sign when dividing by negative.
❌ Mixing up AND/OR.
❌ Using open instead of closed circle incorrectly.
❌ Not testing intervals for correctness.
❌ Leaving answer in inequality when asked for interval notation.

🔹 12. Practice Summary Table

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