Graphing linear inequalities is a key skill in algebra, used to visualize solutions on the coordinate plane. It is essential for solving systems of inequalities, analyzing feasible regions in optimization problems, and understanding variable relationships.
A linear inequality is similar to the linear equation but involves an inequality sign instead of an equals sign. It can be expressed in the form:
ax + by ≤ c, ax + by ≥ c, ax + by < c, ax + by > c
Where:
a, b, and c are constants
x and y are variables.
The graph of the linear inequality represents a region in the coordinate plane where all points satisfy the inequality.
Steps to Graph a Linear Inequality
Step 1: Graph the Corresponding Linear Equation
First, graph the corresponding linear equation. For example for the inequality 2x + 3y ≤ 6 start by graphing the line 2x + 3y = 6.
Plot the Intercepts: Mark the points (3, 0) and (0, 2) on the graph.
Draw the Line: Connect these points with the straight line. Use a solid line for the inequalities involving the ≤ or ≥ and a dashed line for the inequalities involving < or > .
Step 2: Determine the Shading Region
To identify which side of the line to the shade choose a test point that is not on the line often the origin (0, 0) if it is not on the line.
Test Point: Substitute the test point for the inequality.
2(0) + 3(0) ≤ 6 ⟹ 0 ≤ 6 (True)
Since the test point satisfies the inequality shade the region that contains the origin. If the test point does not satisfy the inequality shade the opposite region.
Step 3: Finalize the Graph
Ensure that the entire shaded region represents all the solutions to inequality. Label the graph appropriately showing the inequality and shaded region clearly.
Solved Example of Graphing Linear Inequalities
Example 1: Graphing x − y ≥ 1
Solution :
Graph the Line: For x − y = 1 find the intercepts.
x-intercept: x = 1, y = 0.
y-intercept: x = 0, y = − 1.
Draw a Solid Line: Plot points (1, 0) and (0, − 1) and draw a solid line.
Determine the Shading: Test point (0, 0).
0 − 0 ≥ 1 → 0 ≥ 1 (False). Shade the region opposite to (0, 0).
Example 2: Graphing 2x + 3y < 6
Solution :
Graph the Line: For 2x + 3y = 6 find the intercepts.
x-intercept: x = 3, y = 0.
y-intercept: x = 0, y = 2.
Draw a Dashed Line: The Plot points (3, 0) and (0, 2) and draw a dashed line.
Determine the Shading: Test point (0, 0).
2(0) + 3(0) < 6→0 < 6 (True). Shade the region containing (0,0).
Example 3: Graphing x + 2y ≥ 4
Solution :
Graph the Line: For x + 2y=4 find the intercepts.
x-intercept: x=4, y=0.
y-intercept: x=0, y=2.
Draw a Solid Line: The Plot points (4,0) and (0,2) and draw a solid line.
