Inequalities

1. Concept

• An inequality shows that two expressions are not equal. It uses symbols like < (less than), ≤ (less than or equal to), > (greater than), ≥ (greater than or equal to), and ≠ (not equal).

• The solution set includes all numbers that make the inequality true.

2. Explanations

Solving Basic Inequalities

1. Isolate x: Treat it like an equation, moving terms to get x alone.

2. Multiply/Divide by Positive: The inequality symbol stays the same.

3. Multiply/Divide by Negative: Reverse the inequality symbol (e.g., < becomes >, ≥ becomes ≤).

Graphing on a Number Line

• Draw a number line and mark the boundary value.

• Use an open circle (o) for < or > (boundary not included).

• Use a closed circle (•) for ≤ or ≥ (boundary included).

• Shade the part of the line that makes the inequality true.

  • Example: For x > 5, open circle at 5, shade right.

  • Example: For $x \le -2$, closed circle at -2, shade left.

Compound Inequalities

• "And" (Conjunction): Both parts must be true. The solution is where the individual graphs overlap (intersection).

  • Example: $-4 \le x \le 13$ means x is between -4 and 13 (inclusive).

• "Or" (Disjunction): At least one part must be true. The solution is all values from both graphs combined (union).

3. Formulas

• General Form: $ax + b \triangle c$ (where $\triangle$ is any inequality symbol).

• Solving Rule:

  • If a > 0: $ax + b \triangle c \Rightarrow x \triangle' \frac{c - b}{a}$ (symbol stays the same).

  • If a < 0: $ax + b \triangle c \Rightarrow x \triangle'' \frac{c - b}{a}$ (symbol reverses).

4. Cues

• Remember the Sign Change: Always flip the inequality sign when multiplying or dividing both sides by a negative number.

• Common Outcomes:

  • A specific range (e.g., $x \le 3$).

  • All real numbers (e.g., $0 \le 0$ or 0 < 1 after simplifying).

  • No solution (e.g., 0 < -1 after simplifying).

• Real-world Use: Inequalities help define limits, budgets, or safe ranges.

5. Practice/Example Questions

• Example 1: Solve and graph 2x + 5 > 15

  • Subtract 5: 2x > 10

  • Divide by 2: x > 5

  • Graph: Open circle at 5, shade right.

• Example 2: Solve and graph $-255x - 25 \le 60$

  • Add 25: $-255x \le 85$

  • Divide by -255 (reverse sign): $x \ge -\frac{85}{255} \Rightarrow x \ge -\frac{1}{3}$

  • Graph: Closed circle at $-\frac{1}{3}$, shade right.

• Example 3: Solve and graph the compound inequality: $-4 \le x \le 13$

  • Solution: All numbers from -4 to 13, including -4 and 13.

  • Graph: Closed circle at -4, closed circle at 13, shade between them.

• Test Cases: What are the solution sets for $0 \le 0$ and 0 < 0?