Linear Inequalities

Grade 12 Math - Linear Inequalities

Lesson Objectives

Solve linear inequalities in one and two variables
Graph linear inequalities on a number line and Cartesian plane
Interpret inequality solutions graphically

Lesson Introduction

Linear inequalities describe relationships that are not strictly equal but rather involve "less than," "greater than," and similar comparisons. These inequalities can be solved algebraically and represented visually using number lines (for one variable) or coordinate planes (for two variables).

Core Lesson Content

Solving Linear Inequalities in One Variable

Solve similarly to equations, but reverse the inequality sign when multiplying or dividing by a negative number.

Solving Linear Inequalities in Two Variables

Write in standard form (e.g., $y \leq 2x + 3$ )
Graph the boundary line (dashed if strict, solid if inclusive)
Shade the region satisfying the inequality

Worked Examples

Example 1:
Solve

3x - 5 \leq 10

.
Solution:

3x \leq 15 \Rightarrow x \leq 5

Example 2:
Solve

2 - x > 5

.
Solution:

-x > 3 \Rightarrow x < -3

Example 3:
Graph the inequality

x > -2

on a number line.
Solution: Open circle at -2, arrow to the right.

Example 4:
Solve

-2(x - 1) \geq 4

.
Solution:

-2x + 2 \geq 4 \Rightarrow -2x \geq 2 \Rightarrow x \leq -1

Example 5:
Solve and graph

y < 2x + 1

.
Solution: Graph line

y = 2x + 1

dashed; shade below line.

Example 6:
Solve

4x + 3 \leq 2x + 7

.
Solution:

2x \leq 4 \Rightarrow x \leq 2

Example 7:
Graph

y \geq -x + 2

.
Solution: Graph

y = -x + 2

with a solid line; shade above the line.

Example 8:
A region satisfies

x + y < 4

and

x \geq 0, y \geq 0

. Sketch it.
Solution: Triangle bounded by axes and line

x + y = 4

, excluding the line.

Example 9:
Solve:

3(2x - 1) < 2(3x + 4)

Solution:

6x - 3 < 6x + 8 \Rightarrow -3 < 8

(True for all x)

Example 10:
Graph the system

y \leq x + 2

and

y > -x

.
Solution: Shade the region that lies below

y = x + 2

and above

y = -x

Exercises

Determine whether $x = 2, y = 1$ satisfies $y > x - 3$
[NECO] Solve: $4 - (x + 3) \geq 2x$ [Past Question]
Solve $2(x + 1) > 3x - 4$
[WAEC] Solve $3 - 2x \leq 7$ [Past Question]
Solve and graph $5x - 4 > 11$
Graph the inequality $y \leq |x|$ on the coordinate plane
Find the solution set for $x \geq -4$ on a number line
[NECO] Sketch and label the region that satisfies both $y \geq 2x - 1$ and $y < 4$ [Past Question]
Graph $y < -2x + 5$ on the Cartesian plane
[WAEC] Sketch the region bounded by $y \leq 2x + 1$ and $x \geq 0$ [Past Question]

Conclusion/Recap

Linear inequalities form the basis for analyzing feasible regions in algebra and real-world contexts such as budgeting, production, and optimization. Mastery of solving and graphing them is critical to understanding systems of constraints in mathematics.