Inequalities II

Grade 12 Math - Inequalities

Lesson Objectives

Solve compound and absolute value inequalities algebraically.
Represent solutions of inequalities on number lines.
Graph inequalities on Cartesian planes.
Apply inequalities to contextual word problems.

Lesson Introduction

Inequalities describe a range of values rather than a single solution. Unlike equations, they use symbols such as $<$ , $\leq$ , $>$ , and $\geq$ to indicate the relationship between expressions. This lesson covers solving single-variable and compound inequalities, graphing them on number lines and in coordinate planes, and interpreting real-world problems involving inequalities.

Core Lesson Content

Worked Example

Solving Algebraic Inequalities

To solve inequalities, apply similar rules as equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

Example 1: Solve

3x - 7 > 2

Add 7:

3x > 9

Divide:

x > 3

Example 2: Solve

-2x + 5 \leq 11

Subtract 5:

-2x \leq 6

Divide by -2 and reverse inequality:

x \geq -3

Example 3: Solve

4(2x - 1) > 3x + 5

8x - 4 > 3x + 5 \Rightarrow 5x > 9 \Rightarrow x > \frac{9}{5}

Example 4: Solve

5x + 2 < 3x - 6

Subtract

3x

2x + 2 < -6 \Rightarrow 2x < -8 \Rightarrow x < -4

Example 5: Solve

\frac{x - 2}{3} \geq 1

Multiply both sides by 3:

x - 2 \geq 3 \Rightarrow x \geq 5

Example 6: Solve

\frac{4 - x}{2} < 5

4 - x < 10 \Rightarrow -x < 6 \Rightarrow x > -6

Example 7: Solve

2x - 3 \leq 5x + 6

-3 - 6 \leq 5x - 2x \Rightarrow -9 \leq 3x \Rightarrow x \geq -3

Example 8: Solve

|x - 4| \leq 3

-3 \leq x - 4 \leq 3 \Rightarrow 1 \leq x \leq 7

Example 9: Solve

|2x + 1| > 5

2x + 1 > 5 \Rightarrow x > 2

2x + 1 < -5 \Rightarrow x < -3

Example 10: Solve and graph:

x^2 - 4x < 0

Factor:

x(x - 4) < 0

Critical points:

x = 0, 4

. The solution lies between:

0 < x < 4

Graphing Inequalities on a Number Line

Use open circles for strict inequalities ( $<, >$ ) and closed circles for inclusive inequalities ( $\leq, \geq$ ). Shade the region that satisfies the inequality.

Graphing Inequalities in Two Variables

These are graphed in the Cartesian plane. The boundary line is solid for $\leq, \geq$ and dashed for $<, >$ . Shade the region that satisfies the inequality.

Example 11: Graph

y > 2x - 1

Step 1: Graph the line

y = 2x - 1

with a dashed line.
Step 2: Shade the region above the line.

Example 12: Graph the system:

y \leq x + 2

and

y > -x + 1

Graph both lines, shade where both conditions are satisfied (intersection region).

Exercises

Solve: $2x + 3 < 5x - 6$
[NECO] Solve: $-3(x + 1) \geq 2x - 9$ [Past Question]
Solve and graph: $\frac{x + 4}{2} \leq 3$
[WAEC] Solve: $|x + 2| < 6$ [Past Question]
Solve: $x^2 + x - 6 \geq 0$
[NECO] Solve: $3x - 4 > 2x + 1$ [Past Question]
Solve: $|2x - 3| > 7$
Graph the inequality: $y \leq -x + 4$
Solve and represent on a number line: $x - 5 \geq -2$
[WAEC] A number’s square is less than 16. Find the range of values for the number. [Past Question]

Conclusion/Recap

Inequalities extend the idea of solving equations by identifying value ranges. Algebraic manipulation, understanding of absolute values, and graphing skills are crucial. With regular practice, you can interpret and solve increasingly complex inequality problems in both pure and applied contexts.