PSAT/NMSQT Math: Systems of Inequalities
Systems of inequalities questions on the PSAT/NMSQT are almost always graph-based: instead of solving algebraically, you read, interpret, or reverse-engineer shaded regions on a coordinate plane. This free, complete lesson covers graphing a single linear inequality, testing whether a point is a solution, describing the overlapping solution region of a full system, and working backward from a graph to identify the inequalities that produced it. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Graphing a Single Linear Inequality
To graph a linear inequality, first graph the boundary line as if it were an equation. Use a dashed line for strict inequalities (< or >), since points on the line itself are not included, and a solid line for ≤ or ≥, since the boundary points are included. Then shade the side of the line containing the solutions.
Worked Example: Describe how to graph y < 2x + 3.
Graph y = 2x + 3 as a dashed line (strict inequality).
Shade below the line, since y is less than the line's value at every x
1. When graphing y ≥ −x + 5, what type of boundary line should be drawn?
Explanation: The symbol ≥ includes equality, so points exactly on the line are solutions, requiring a solid boundary line.
2. For the inequality y > 3x − 2, which side of the boundary line should be shaded?
Explanation: Since y is greater than the boundary expression, all points above the line satisfy the inequality.
3. What type of boundary line and shading direction describes x ≤ 4 on a standard xy-plane?
Explanation: x = 4 is a vertical line; since x ≤ 4 includes equality, the line is solid, and all points with x-values less than or equal to 4 (to the left) are shaded.
2. Testing Whether a Point Is a Solution
To check whether a specific point is a solution to an inequality (or a system of inequalities), substitute its x- and y-coordinates directly into each inequality. If the resulting statement is true for every inequality in the system, the point is a solution.
Worked Example: Is the point (1, 4) a solution to y ≤ 2x + 5?
Substitute x = 1, y = 4:
4 ≤ 2(1) + 5 = 7, which is true, so (1, 4) is a solution
1. Is the point (2, 3) a solution to y > x + 4?
Explanation: Substituting: 3 > 2 + 4 = 6 is false, so (2, 3) is not a solution.
2. Which point is a solution to the system y ≤ x + 1 and y ≥ −x + 1?
Explanation: For (2, 1): 1 ≤ 3 is true, and 1 ≥ −1 is true, so both inequalities hold. The other points each fail at least one inequality.
3. Is the point (−1, −1) a solution to the system x ≥ 0 and y ≤ 2x + 3?
Explanation: A point must satisfy every inequality in the system. Since x = −1 does not satisfy x ≥ 0, the point already fails, regardless of the second inequality.
3. The Overlapping Solution Region of a System
A system of inequalities graphs each inequality's shaded region separately, and the solution to the whole system is the overlap, the region shaded by every inequality at once. Any point inside the overlapping region satisfies all the inequalities simultaneously.
Worked Example: A graph shows y ≤ 5 shaded below a horizontal line, and x ≥ 2 shaded to the right of a vertical line. Describe the system's solution region.
The solution is the region below y = 5 AND to the right of x = 2, forming a shared corner region
1. A graph shows the overlapping shaded region of a system bounded above by a solid horizontal line at y = 6 and to the left by a solid vertical line at x = 1. Which system produces this overlap?
Explanation: The overlap sits below y = 6 (y ≤ 6) and to the right of x = 1 (x ≥ 1), matching the described bounded corner region.
2. Two inequalities' shaded regions on a graph do not overlap anywhere. What does this tell you about the system?
Explanation: A system's solution is the set of points satisfying every inequality at once; if the shaded regions never overlap, no point satisfies both, so there is no solution.
3. A budget system requires x ≥ 0, y ≥ 0, and x + y ≤ 10, where x and y represent quantities purchased. What kind of region does this describe?
Explanation: x ≥ 0 and y ≥ 0 restrict the region to the first quadrant, and x + y ≤ 10 caps it with a diagonal boundary, forming a closed triangular region.
4. Reverse-Engineering a System from Its Graph
Some PSAT questions show a completed graph with a shaded region and ask you to identify which system of inequalities produced it. Work one boundary line at a time: find its equation, determine whether the line is solid or dashed, and use the shaded direction to decide the inequality symbol.
Worked Example: A graph shows a dashed line through (0, 2) with slope 1, shaded above the line. Write the inequality.
The boundary line's equation is y = x + 2. Dashed means strict inequality, and shading above means "greater than":
y > x + 2
1. A graph shows a solid line through (0, −1) with slope 2, shaded below the line. Which inequality matches this graph?
Explanation: The boundary line is y = 2x − 1. Solid means ≤ or ≥, and shading below means "less than or equal to," giving y ≤ 2x − 1.
2. A graph shows a dashed vertical line at x = 3, shaded to the right. Which inequality matches this graph?
Explanation: Dashed means strict inequality, and shading to the right of x = 3 means values greater than 3, giving x > 3.
3. A graph shows two boundary lines: a solid line through (0, 4) with slope −1, shaded below, and a solid horizontal line at y = 0, shaded above. Which system matches this graph?
Explanation: The first line, y = −x + 4, is shaded below, giving y ≤ −x + 4. The line y = 0 is shaded above, giving y ≥ 0.
Common Mistakes to Avoid
- Mixing up solid and dashed boundary lines. Solid means the inequality includes equality (≤ or ≥); dashed means it's strict (< or >) and the boundary itself is excluded.
- Shading the wrong side of the boundary line. Always test a point not on the line (the origin is often easiest) in the original inequality to confirm which side to shade.
- Only checking one inequality when testing a point against a system. A point must satisfy every inequality in the system to be a valid solution.
- Assuming a system with two inequalities always has a bounded, closed solution region. Many systems produce an unbounded region that extends infinitely in one or more directions.
- Misreading which variable a vertical or horizontal boundary line restricts. A vertical line restricts x; a horizontal line restricts y.
Frequently Asked Questions
What's the fastest way to figure out which side of a line to shade?
Pick a test point that is clearly not on the boundary line, often (0, 0) if the line doesn't pass through the origin, and substitute it into the original inequality. If the statement is true, shade the side containing that point; if false, shade the other side.
Can a system of inequalities have no solution?
Yes. If the shaded regions of the individual inequalities never overlap anywhere on the graph, there is no point that satisfies every inequality, so the system has no solution.
Where can I practice more problems like these?
The Systems of Inequalities quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.