Learn linear inequalities in two variables
A linear inequality in two variables divides the coordinate plane into a shaded solution half-plane. The boundary is solid for ≤ or ≥ and dashed for < or >.
You will graph a boundary line, choose solid or dashed, and shade the correct side.
The idea in plain English
A two-variable inequality has many ordered-pair solutions. They fill one whole side of a boundary line.
The boundary shows where the two sides are equal. The inequality tells whether that boundary counts and which side to shade.
How to recognise this kind of question
- Both x and y appear with an inequality sign.
- The answer is a shaded region on a coordinate plane.
- You must decide between a solid and dashed line.
Quick decision: Draw the equality line; use solid for ≤ or ≥, dashed for < or >; then shade the solution side.
The method
- Rewrite the boundary in slope-intercept form when needed.
- Draw a solid or dashed boundary line based on the inequality symbol.
- Shade above for y > or y ≥, and below for y < or y ≤.
- Use a test point when the inequality is not already solved for y.
Worked examples
- Shade below the boundary because y is less than the line's value.
Answer: Dashed boundary; shade below
Graph y ≥ −x + 1.
- Draw y = −x + 1 as a solid line because equality is included.
- Shade above the line because y is greater.
Answer: Solid boundary; shade above
Your turn—with help
Graph y > 2x − 3.
Need a hint?
Strict > means dashed. Greater y-values are above.
Show the answer and steps
- Graph y = 2x − 3.
- Make the line dashed.
- Shade above it.
Answer: Dashed boundary; shade above
Watch out for this mistake
Incorrect: y < x + 2 uses a solid line
Why it fails: The strict symbol < does not include points on the boundary.
Do this instead: Use a dashed boundary and shade below.
Quick check
Try these without looking. Then open each answer to check your thinking.
y ≤ x + 4
Need a hint?
≤ includes the boundary and means below.
Show the answer and steps
- Draw a solid line.
- Shade below.
Answer: Solid boundary; shade below
y < −2
Need a hint?
This boundary is horizontal.
Show the answer and steps
- Draw dashed y = −2.
- Shade lower y-values.
Answer: Dashed line y = −2; shade below
Does (0, 0) satisfy y ≥ x − 1?
Need a hint?
Substitute x = 0 and y = 0.
Show the answer and steps
- 0 ≥ 0 − 1 becomes 0 ≥ −1.
- That statement is true.
Answer: Yes
Common mistakes
- Using a solid boundary for a strict inequality.
- Shading above when the inequality says y is less than the boundary.
- Changing the slope while converting the equation.
Practice tips
- Circle the inequality symbol before drawing the boundary.
- Test (0, 0) when it is not on the boundary.