Linear Inequalities in Two Variables
How to Shade the region of linear inequalities
Use this to draw and shade the region that satisfies one or more linear inequalities.
Before you start
- Plotting a straight line from its equation
- Reading the symbols ≤, ≥, < and >
- Substituting a point into an inequality
When to use it
Use this to draw and shade the region that satisfies one or more linear inequalities.
The steps
- Draw each boundary line: solid for ≤ or ≥, dashed for < or >.
- Pick a test point not on the line, usually the origin (0, 0).
- Substitute it into the inequality to see if it is satisfied.
- Shade the side that satisfies the inequality.
- For several inequalities, the answer is where all shaded regions overlap.
Worked example
Shade the region that satisfies y ≥ x + 1.
- Draw the boundary line y = x + 1. Because the symbol is ≥, draw it as a solid line.
- Pick a test point not on the line; the origin (0, 0) works since 0 ≠ 0 + 1.
- Substitute (0, 0): 0 ≥ 0 + 1 gives 0 ≥ 1, which is false.
- Since the origin fails, shade the other side, the region above the line, away from (0, 0).
- There is only one inequality, so that shaded region is the answer.
A second example, with a twist
There are two inequalities, one solid and one dashed, so the answer is the overlap of both shaded regions. Shade the region that satisfies both x + y ≤ 4 and x > 1.
- Draw both boundaries: x + y = 4 as a solid line (because of ≤) and x = 1 as a dashed line (because of >).
- Use the test point (0, 0), which is on neither line.
- Substitute into each: for x + y ≤ 4, 0 ≤ 4 is true; for x > 1, 0 > 1 is false.
- Shade to keep the origin side of x + y = 4, and shade the far side of x = 1 (to its right), since the origin failed there.
- The answer is where the two shadings overlap: to the right of the dashed line x = 1 and on or below the solid line x + y = 4.
Formula pages
Practise this in a KBAT problem
Frequently asked questions
Solid line or dashed line, how do I decide?
Look at the inequality symbol. Use a solid line for ≤ or ≥, because points on the line are included.
Use a dashed line for < or >, because points on the line are not included. The line style is a quick signal of whether the boundary itself counts.
Why is (0, 0) the usual test point?
The origin makes the arithmetic easy, because multiplying and adding zeros is quick. It only fails as a test point when the boundary line passes through the origin itself; then you must pick any other point not on the line, such as (1, 0), and test that instead.
For several inequalities, which part is the answer?
The answer is the region where every inequality is satisfied at once, the overlap of all the shaded areas. A point in the final region must obey all the inequalities together.
If you shade each one lightly, the darkest part where they all cross is the region you want.