Algebra
Single-variable inequalities
Linear inequalities
20 practice questions
1 video lesson
Theory + worked examples
Practice
20 questions
Practice questions
Every question with a fully worked solution.
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1 video(s)
- Algebra 1 Chapter 5: Solving and Graphing Linear Inequalities Watch
Theory
A linear inequality in two variables graphs as a half-plane:
- Graph the boundary line (\(y=mx+b\)).
- Make it dashed for \(<,>\) or solid for \(\le,\ge\).
- Test a point (often \((0,0)\)) not on the line.
- Shade the side that makes the inequality true.
A true test point means shade its side; a false one means shade the other.
A half-plane above the boundary line.
Graphing a two-variable inequality.
Boundary and shading:
\[y>mx+b:\ \text{dashed, shade above};\quad y<mx+b:\ \text{shade below}\]
Use a test point to be sure which side to shade.
How to graph a linear inequality
- Graph the boundary line.
- Dashed for strict, solid for inclusive.
- Test a point off the line.
- Shade the side that satisfies the inequality.
Example 1 — Solid or dashed?
For \(y\le x+2\), is the boundary solid or dashed?
Solution
\(\le\) includes the line, so it is solid.
| \(\le\) | \(\Rightarrow\) | \(\text{solid line}\) |
Example 2 — Test a point
For \(y>2x-1\), does \((0,0)\) satisfy it?
Solution
Substitute \((0,0)\).
| \(0\) | \(>\) | \(2(0)-1\) |
| \(0\) | \(>\) | \(-1\ \checkmark\) |
Yes — shade the side containing \((0,0)\).
Example 3 — Which side to shade
For \(y\ge x\), which side of the line is shaded?
Solution
\(y\ge x\) means \(y\) is above the line, so shade above.
Example 4 — A point not in the region
Is \((3,0)\) a solution of \(y>2x-1\)?
Solution
Check \((3,0)\).
| \(0\) | \(>\) | \(2(3)-1=5?\) |
| \(0\) | \(>\) | \(5\ \text{false}\) |
No — \((3,0)\) is not in the region.
Common pitfalls
Dashed for \(<,>\); solid for \(\le,\ge\).
Test a point — don't guess the side.
\((0,0)\) is easiest unless it lies on the line.
Frequently asked questions
How do you graph a two-variable linear inequality?
Draw the boundary line, test a point, and shade the correct side.
When is the boundary dashed?
For strict inequalities \(<\) and \(>\).
How do you decide which side to shade?
Test a point; shade the side that makes the inequality true.
What is the solution to a linear inequality?
Every point in the shaded half-plane.
More in Single-variable inequalities