We are going to begin this section by reviewing the technique for graphing inequalities in two variables.
Click here for a presentation on Graphing Inequalities in 2 Variables
Now that we have looked at the process, let’s write down some terms and steps. We have been dividing the coordinate plane into half-planes when we create a boundary, by using a line or a broken line, and then shading above or below it.
Half-Planes
A boundary line divides the coordinate plane into two regions called half-planes.
The techniques used in the presentation to graph inequalities involve using a solid line or a broken line for the boundary. Then the portion of the graph where the solutions occur is shaded.
Steps for Graphing an Inequality
Solve the inequality in terms of y.
Graph the point for the y-intercept.
Graph the rise and run for the slope.
Draw a boundary line using this guideline:
If > or <, use a broken line (dashed line).
If ≥ or ≤, use a solid line.
Shade the half-plane using this guideline:
If > or ≥, shade above the boundary.
If < or ≤, shade below the boundary.
You can check by picking a point in the shaded area and substituting into the original inequality.
Example 1: Graph an Inequality
Graph and check: 2x + y > 5.
Solve in terms of y.
2x + y > 5
y > 5 – 2x
Graph the y-intercept of 5.
Graph the slope of -2/1.
Draw a boundary of a broken line.
Shade above the broken line.
Check using (3, 1), which is in the shaded portion.
2x + y > 5
2(3) + 1 > 5
7 > 5 √
Example 2: Graph an Inequality
Graph and check: 3y – 4x ≤ -12
Solve in terms of y.
3y – 4x ≤ -12
3y ≤ -12 + 4x
y ≤ -4 + 4/3 x
Graph the y-intercept of -4.
Graph the slope of 4/3.
Draw a boundary of a solid line.
Shade below the solid line.
Check using (4, 1), which is within the shaded portion.
3y – 4x ≤ -12
3(1) – 4(4) ≤ -12
-13 ≤ -12 √
Inequalities Involving Horizontal Boundary Lines
To graph y ≥ a, draw a horizontal line boundary and shade above.
To graph y ≤ a, draw a horizontal line boundary and shade below.
Example 3: Special Graph
Graph and check: y > -2.
The inequality is already solved in terms of y.
The slope is zero.
The y intercept is -2.
This will be a horizontal broken line cutting across -2 on the y axis.
Then we will shade above the line where the y values of the points are all greater than -2.
Check that the point (2, 1) is in the shaded area.
y > -2
1 > -2 √
Inequalities Involving Vertical Boundary Lines
To graph x ≥ a, draw a vertical line boundary and shade to the right.
To graph x ≤ a, draw a vertical line boundary and shade to the left.
Example 4: Vertical Line Boundary
Graph x ≥ 3.
x = 3 is a vertical line graph.
Shade to the right, where the x values of the points are all greater than three.
Check by substituting (5, 2) from the shaded area.
x ≥ 3
5 ≥ 3 √
Half-Planes
Click here for a presentation on Graphing Inequalities in 2 Variables
