Which graph shows the solution to the system of linear inequalities?
The correct graph is the one where the shaded regions of both inequalities overlap. That darker, double-shaded area is the solution. Use solid boundary lines for ≤ or ≥ and dashed lines for < or >.
The answer
The solution to a system of linear inequalities is the set of every point (x, y) that satisfies all of the inequalities at the same time. On a graph, each inequality shades one half-plane. Where those shaded half-planes overlap — the region covered by both — is the solution set. So the correct graph is the one whose double-shaded (darker) region shows that overlap, bounded correctly by solid or dashed lines.
Two details decide whether a graph is right:
- Boundary style. Draw a solid line when the inequality includes equality (≤ or ≥), because points on the line are part of the solution. Draw a dashed line for strict inequalities (< or >), because the line itself is excluded.
- Shading direction. Pick a test point not on the line — (0, 0) is easiest when the line doesn't pass through the origin. Plug it in; if the inequality is true, shade the side containing that point, otherwise shade the other side.
Why the other options are wrong
Most incorrect graphs fail in one of three predictable ways, and knowing them lets you eliminate distractors fast:
- Only one region shaded. A graph that shades just one inequality's half-plane represents a single inequality, not the system. The system requires the intersection, so a single shaded band cannot be the answer.
- The wrong side shaded. If a graph shades above a line when the test point shows it should be below (or vice versa), the overlap lands in the wrong place. Always re-check with a test point.
- Wrong line style. A graph that uses a solid line for a strict
<or a dashed line for≤misrepresents whether the boundary is included. Even with correct shading, the wrong line type makes the graph wrong.
A graph where the two shaded regions do not overlap at all would represent a system with no solution — valid only if the inequalities are genuinely contradictory (for example, parallel lines shaded away from each other).
The bigger picture
Think of each inequality as a filter. The first inequality throws away every point on one side of its line; the second throws away points on one side of its line. Whatever survives both filters is your answer. That is exactly why the overlap — and only the overlap — counts.
To verify any candidate graph, grab a point clearly inside the double-shaded region and substitute it into both original inequalities. If both come out true, the graph is correct. If even one is false, the shading is wrong. This test-point method is more reliable than eyeballing, and it works no matter how many inequalities the system contains. Corner points where the boundaries meet also matter in optimization (linear programming), where the best solution always sits at a vertex of the overlap region.
- 1
Graph each boundary line
Solid line for ≤ or ≥ (points on the line count); dashed line for < or > (line excluded).
- 2
Test a point for each inequality
Use (0,0) if the line misses the origin. Substitute it; if true, shade that side, if false shade the other.
- 3
Find the overlap
The region shaded by BOTH inequalities — the darker double-shaded area — is the solution set.
- 4
Verify with a point inside
Pick a point in the overlap and plug it into both original inequalities. Both must be true.
Frequently asked
How do you find the solution region of a system of linear inequalities?
Graph and shade each inequality separately, then identify where the shaded regions overlap. That overlapping area contains every point that satisfies all inequalities at once, so it is the solution region.
When do you use a solid line vs a dashed line?
Use a solid line when the inequality includes equality (≤ or ≥), because points on the line are solutions. Use a dashed line for strict inequalities (< or >), because the boundary line itself is not included.
How do you know which side of the line to shade?
Choose a test point off the line, often (0, 0), and substitute it into the inequality. If the statement is true, shade the side containing that point; if false, shade the opposite side.
What does the overlapping region of two inequalities represent?
It represents all the ordered pairs that satisfy both inequalities simultaneously. Any point inside the overlap makes every inequality in the system true, which is exactly the definition of the system's solution.