Algebra
Systems of equations and inequalities
Systems of linear inequalities
20 practice questions
2 video lessons
Theory + worked examples
Theory
A system of linear inequalities is solved graphically:
- Graph each inequality and shade its half-plane.
- The solution is the overlap of the shaded regions.
- Any point in the overlap satisfies all the inequalities.
Dashed boundaries for \(<,>\); solid for \(\le,\ge\).
The overlap of the shaded regions is the solution.
Solving a system of inequalities.
The solution set:
\[\text{solution}=\text{intersection of the half-planes}\]
Test a point in the overlap to confirm.
How to solve a system of inequalities
- Graph the first inequality and shade.
- Graph the second and shade.
- Find where the shadings overlap.
- Test a point in the overlap.
Example 1 — The overlap
What is the solution region of a system of two inequalities?
Solution
The overlap of the two shaded half-planes.
Example 2 — Test a point
Does \((1,3)\) satisfy \(y\ge x\) and \(y\le -x+5\)?
Solution
Check both.
| \(3\ge1\) | \(\checkmark\) | |
| \(3\le4\) | \(\checkmark\) |
Yes — it is in the overlap.
Example 3 — Boundary style
Are the boundaries solid or dashed for \(\ge\) and \(\le\)?
Solution
Inclusive inequalities use solid boundary lines.
Example 4 — No overlap
What if the shaded regions don't overlap?
Solution
Then the system has no solution.
Common pitfalls
The solution is the overlap, not either region alone.
Dashed vs solid boundaries follow the inequality.
No overlap means no solution.
Frequently asked questions
How do you solve a system of inequalities?
Graph and shade each, then take the overlap.
What is the solution region?
The overlap of the shaded half-planes.
When are boundaries dashed?
For strict inequalities \(<\) and \(>\).
What if the regions don't overlap?
There is no solution.
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