Inequalities

Lesson Objectives

By the end of this lesson, you should be able to:

Understand the meaning and symbols of inequalities.
Solve inequalities involving linear expressions.
Apply inverse operations and rules of inequalities.
Interpret and graph solutions on a number line.

Lesson Introduction

Inequalities are mathematical statements that compare two expressions using inequality signs: less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). Solving inequalities involves finding the values that make the statement true, just like solving equations — but with special rules for negative multiplication or division.

Examples

Understanding Inequality Symbols

Example: Interpret the inequality $5 < 7$ .

Answer: 5 is less than 7.

Example: Interpret the inequality $8 \geq 3$ .

Answer: 8 is greater than or equal to 3.

Combining Like Terms

Example: Solve $2x + 3x - 5 > 10$ .

Combine like terms: $5x - 5 > 10$
Add 5 to both sides: $5x > 15$
Divide by 5: $x > 3$

Example: Solve $4x + 6x - 8 \leq 12$ .

Combine: $10x - 8 \leq 12$
Add 8: $10x \leq 20$
Divide by 10: $x \leq 2$

Isolating the Variable

Example: Solve $3x - 7 \leq 11$ .

Add 7: $3x \leq 18$
Divide: $x \leq 6$

Example: Solve $2x + 9 > 17$ .

Subtract 9: $2x > 8$
Divide: $x > 4$

Solving Inequalities with Fractions

Example: Solve $\frac{1}{2x} + \frac{3}{4} < \frac{5}{8}$ .

Subtract: $\frac{1}{2x} < \frac{5}{8} - \frac{3}{4} = \frac{-1}{8}$
Multiply: $-2x < 8$ ⇒ $x > -4$

Example: Solve $\frac{x + 2}{3} \geq \frac{x - 1}{2}$ .

Cross-multiply: $2(x + 2) \geq 3(x - 1)$
Expand: $2x + 4 \geq 3x - 3$
Solve: $7 \geq x$ ⇒ $x \leq 7$

Solving Inequalities with Parentheses

Example: Solve $2(x + 3) - 4 > 11$ .

Expand: $2x + 6 - 4 > 11$
Simplify: $2x + 2 > 11$
Solve: $x > 4.5$

Example: Solve $-3(x - 2) \leq 9$ .

Expand: $-3x + 6 \leq 9$
Subtract: $-3x \leq 3$ ⇒ $x \geq -1$

Multiplying or Dividing by a Negative Number

Example: Solve $-2x > 6$ .

Divide: $x < -3$ (Flip the sign)

Example: Solve $-5x \leq 15$ .

Divide: $x \geq -3$

Graphing Inequalities

Example: Graph $x \leq 3$ on a number line.

Answer: Place a closed circle at 3 and shade to the left.

Example: Graph $x > -2$ .

Answer: Place an open circle at -2 and shade to the right.

Applications and Word Problems

Example: A store offers a 20% discount. If you want to spend less than ₦50, what’s the max original price?

Let $x$ = original price
$0.8x < 50$ ⇒ $x < 62.50$

Example: A bus carries at most 60 people. If 5 people are already on board, how many more can enter?

Let $x$ be the number of people who can still enter.
$x + 5 \leq 60$ ⇒ $x \leq 55$

Exercises

[JSCE] Solve $0.8x + 10 \leq 50$ . (Past Question)
Solve $2(x + 3) - 5 > 11$ .
[WAEC] Solve $4x - 3 > 5$ . (Past Question)
Solve $5x - 7 < 3x + 1$ .
Solve $\frac{x + 1}{2} \geq \frac{x - 3}{3}$ .
[NECO] Solve $2(x - 3) < 8$ . (Past Question)
Solve $2(x + 4) - 3x < 7$ .
Solve $-3x + 2 > -4$ .
[NABTEB] Solve $-2(x - 1) \leq 6$ . (Past Question)
Solve $\frac{2x - 1}{5} < \frac{3x + 2}{10}$ .

Conclusion/Recap

Inequalities allow us to express a range of possible solutions instead of just one. Always remember to reverse the inequality sign when multiplying or dividing by a negative number. Practice solving different types — linear, fractions, and word problems — to master this topic.