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.