Inequalities

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Inequalities — Quick Facts

Basic Inequality Signs:

• $<$ : less than
• $>$ : greater than
• $\leq$ : less than or equal to
• $\geq$ : greater than or equal to
• $\neq$ : not equal to

Properties of Inequalities:

• If $a < b$, then $a + c < b + c$ (adding same value to both sides)
• If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$ (dividing by positive)
• If $a < b$ and $c < 0$, then $\frac{a}{c} > \frac{b}{c}$ (dividing by negative flips sign)
• If $a < b$ and both positive, then $\frac{1}{a} > \frac{1}{b}$ (reciprocals flip)

Number Line Representation:

• $x > 3$: open circle at 3, shade right
• $x \geq 3$: closed circle at 3, shade right
• $-2 < x \leq 5$: open circle at -2, closed at 5, shade between

⚡ CAT Exam Tip: When multiplying or dividing an inequality by a negative number, ALWAYS flip the inequality sign.

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Inequalities — Study Guide

Solving Linear Inequalities:

Example: Solve $3x - 2 < 7x + 6$

$3x - 7x < 6 + 2$ $-4x < 8$ $x > -2$ (flipped sign because dividing by -4)

Compound Inequalities:

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

Add 1: $-2 \leq 2x < 8$ Divide by 2: $-1 \leq x < 4$

In interval notation: $[-1, 4)$

Quadratic Inequalities:

To solve $x^2 - 5x + 6 < 0$:

1. Find roots: $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$
2. The parabola $y = x^2 - 5x + 6$ opens upward
3. The expression is negative between the roots

Solution: $2 < x < 3$ or $(2, 3)$

Example: Solve $x^2 + 4x + 3 \geq 0$

$(x+1)(x+3) \geq 0$

Critical points: $x = -1$ and $x = -3$ Sign chart: Test $x < -3$, $-3 < x < -1$, $x > -1$

• $x < -3$: both factors negative → positive ≥ 0 ✓
• $-3 < x < -1$: (negative)(positive) → negative ✗
• $x > -1$: both positive → positive ≥ 0 ✓

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

⚡ Common Student Mistake: Forgetting that quadratic inequalities have TWO critical points (the roots), not one. Drawing the sign chart helps avoid this.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Inequalities — Comprehensive Notes

Absolute Value Inequalities:

Type 1: $|x| < a$ (where $a > 0$) Solution: $-a < x < a$

Type 2: $|x| > a$ (where $a > 0$) Solution: $x < -a$ or $x > a$

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

$-5 < 2x - 3 < 5$ Add 3: $-2 < 2x < 8$ Divide by 2: $-1 < x < 4$

Example: Solve $|x + 1| > 3$

$x + 1 > 3$ or $x + 1 < -3$ $x > 2$ or $x < -4$

Rational Inequalities:

Example: Solve $\frac{x-2}{x+3} > 0$

Critical points: $x = 2$ (zero of numerator) and $x = -3$ (undefined)

Sign chart:

• $x < -3$: $\frac{(-)}{(-)} = positive$ ✓
• $-3 < x < 2$: $\frac{(-)}{(+)}= negative$ ✗
• $x > 2$: $\frac{(+)}{(+)}= positive$ ✓

Note: $x = 2$ makes numerator 0 → value is 0 (not > 0), so exclude $x = 2$ from solution. $x = -3$ makes denominator 0 → undefined.

Solution: $(-\infty, -3) \cup (2, \infty)$

Wavy Curve Method:

For $\frac{(x-1)(x-2)}{(x-3)(x+1)} \geq 0$:

1. Find all critical points (zeros and undefined points): $x = 1, 2, 3, -1$
2. Plot on number line
3. Starting from rightmost point, alternate signs (since all factors are linear)
4. Include zeros where inequality permits (≥ includes, > excludes)

Inequalities with Modulus and Quadratic Combined:

Example: $|x^2 - 4| \leq 3$

$-3 \leq x^2 - 4 \leq 3$ Add 4: $1 \leq x^2 \leq 7$ $\sqrt{1} \leq |x| \leq \sqrt{7}$ So $|x| \geq 1$ and $|x| \leq \sqrt{7}$

Solution: $x \in [-\sqrt{7}, -1] \cup [1, \sqrt{7}]$

Min/Max Problems Using Inequalities:

For real numbers $x, y$ with $x + y = k$, the product $xy$ is maximised when $x = y = k/2$, giving maximum product $k^2/4$.

AM-GM Inequality: $$\frac{x + y}{2} \geq \sqrt{xy}$$ Equality when $x = y$.

Cauchy-Schwarz Inequality: $$(a_1^2 + a_2^2)(b_1^2 + b_2^2) \geq (a_1 b_1 + a_2 b_2)^2$$

JAMB Pattern Analysis (CAT 2015-2024):

• 2015: Linear inequality solving
• 2017: Compound inequality
• 2019: Quadratic inequality using sign chart
• 2021: Absolute value inequality
• 2023: Rational inequality with wavy curve
• 2024: Mixed — quadratic + absolute value combined

⚡ Exam Strategy: For polynomial/rational inequalities, always use the sign chart method. Identify critical points (where expression = 0 or is undefined), then test intervals.