Systems of Linear Inequalities: Free SAT Math Lesson
A system of linear inequalities combines two skills you've already built, solving systems of equations and reading two-variable inequalities, into one: finding the region of points that satisfies every inequality in the system at once. This free lesson covers what makes a point a solution to a system, how to test a point against multiple inequalities, how to find the intersection point of the boundary lines, how to read an overlapping solution region, and how systems of inequalities model real-world constraints. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. Solutions to a System of Inequalities
A system of inequalities is a set of two or more inequalities with the same variables. An ordered pair is a solution to the system only if it satisfies every inequality in the system at the same time, not just one of them. Graphically, the solution set is the overlap, the region where the individual shaded half-planes intersect.
Worked Example: Is (2, 1) a solution to the system y < 3x and y > −x + 1?
1 < 3(2) = 6? Yes.
1 > −(2) + 1 = −1? Yes.
Since (2, 1) satisfies both inequalities, it is a solution to the system.
1. Is (2, 1) a solution to the system y < 3x and y > −x + 1?
Explanation: This is the worked example above: both 1 < 6 and 1 > −1 are true, so (2, 1) satisfies both inequalities.
2. Is (0, 0) a solution to the system y ≥ 2x − 3 and y ≤ −x + 4?
Explanation: First inequality: 0 ≥ 2(0) − 3 = −3, true. Second inequality: 0 ≤ −(0) + 4 = 4, true. Both hold, so (0, 0) is a solution.
3. Is (3, 5) a solution to the system y < x + 1 and y > 2x − 4?
Explanation: First inequality: is 5 < 3 + 1 = 4? No, this fails. Since a point must satisfy every inequality in the system, (3, 5) is not a solution. It's worth noting the second inequality does hold (5 > 2), but that alone isn't enough.
4. For a point to be a solution to a system of inequalities, it must satisfy:
Explanation: A system requires every inequality to be true at once for a given point, that's what makes the solution set the overlap of the individual regions rather than their combination.
5. A system consists of x ≥ 0 and y ≥ 0. Which of the following best describes the solution set?
Explanation: x ≥ 0 includes all points on or to the right of the y-axis, and y ≥ 0 includes all points on or above the x-axis. Together, that's all of Quadrant I, plus the two boundary axes themselves, since both symbols include equality.
2. Intersection Points and Testing Regions
The boundary lines of a system often cross at a point, called the intersection point, which you can find using substitution or elimination, exactly like solving a system of equations. This point is useful because it's the corner where the shaded regions of the individual inequalities meet.
Worked Example: Find the intersection point of the boundary lines for the system y ≤ 2x + 1 and y ≥ −x + 4.
2x + 1 = −x + 4
3x = 3, so x = 1
y = 2(1) + 1 = 3
The boundary lines intersect at (1, 3).
1. What is the intersection point of the boundary lines y = 3x − 2 and y = −x + 6?
Explanation: Set the equations equal: 3x − 2 = −x + 6, so 4x = 8, and x = 2. Then y = 3(2) − 2 = 4. The intersection point is (2, 4).
2. A system consists of y ≤ x + 3 and y ≥ 2x − 1. Which point lies in the overlapping solution region?
Explanation: Testing (0, 0): 0 ≤ 0 + 3 = 3, true, and 0 ≥ 2(0) − 1 = −1, true. Both hold. The other points each fail at least one inequality: (3, 7) fails since 7 ≤ 6 is false; (0, 5) fails since 5 ≤ 3 is false; (−2, −6) fails since −6 ≥ −5 is false.
3. For the system y > x − 2 and y < −2x + 8, is the point (3, 2) in the solution region?
Explanation: First inequality: 2 > 3 − 2 = 1, true. Second inequality: is 2 < −2(3) + 8 = 2? No, since 2 is not strictly less than 2 (the point lies exactly on this boundary line, and the inequality is strict). So (3, 2) fails the second inequality.
4. In a system of two linear inequalities, the graph of the solution set is the region where:
Explanation: The solution set of a system is only where every individual inequality's shaded region overlaps, not where just one of them is shaded.
5. Two boundary lines, y = x + 4 and y = −x + 4, intersect at which point?
Explanation: Set the equations equal: x + 4 = −x + 4, so 2x = 0, and x = 0. Then y = 0 + 4 = 4. The intersection point is (0, 4).
3. Reading Graphs and Real-World Constraints
To read a graph of a system, identify each boundary line's slope, y-intercept, and whether it's solid or dashed, then determine each one's shaded side, exactly as in the previous lesson, and combine them. Systems of inequalities also commonly model real-world situations with more than one limit or requirement at once, such as a budget constraint combined with a minimum quantity.
Worked Example: A graph shows a solid line through (0, 3) and (3, 0), shaded toward the origin, together with a dashed line through (0, −2) with slope 1, shaded above the line. Write the system of inequalities this graph represents.
The first line has slope (0 − 3)/(3 − 0) = −1 and y-intercept 3, so its equation is y = −x + 3. It's solid and shaded toward the origin, where 0 ≤ 3 is true, giving y ≤ −x + 3.
The second line has slope 1 and y-intercept −2, so its equation is y = x − 2. It's dashed and shaded above, giving y > x − 2.
The system is y ≤ −x + 3 and y > x − 2.
1. A graph shows a solid line through (0, 3) and (3, 0), shaded toward the origin, together with a dashed line through (0, −2) with slope 1, shaded above the line. Which system does this graph represent?
Explanation: This is the worked example above: y ≤ −x + 3 (solid, toward origin) and y > x − 2 (dashed, shaded above).
2. A solution region is described by y ≤ 5, y ≥ 0, and x ≥ 0 all holding at once. Which point lies in this region?
Explanation: (2, 3) satisfies all three: 3 ≤ 5, 3 ≥ 0, and 2 ≥ 0. The others each fail one condition: (2, 6) has y > 5; (−1, 2) has x < 0; (2, −3) has y < 0.
3. Which system of inequalities has an empty solution set, meaning no point satisfies both inequalities?
Explanation: In choice A, satisfying both would require y to be both greater than x + 5 and less than x − 3 at the same time. But x − 3 is always less than x + 5, so no value of y can fit between them, the boundary lines are parallel and face away from each other, leaving no overlap.
4. A company sells x units of Product A and y units of Product B. Production requires x + y ≤ 50 and y ≥ 10. Which production plan satisfies both constraints?
Explanation: For x = 30, y = 15: 30 + 15 = 45 ≤ 50, true, and 15 ≥ 10, true. Both hold. The others fail: (45, 15) gives 60 ≤ 50, false; (30, 5) gives 5 ≥ 10, false; (60, 10) gives 70 ≤ 50, false.
Common Mistakes to Avoid
- Accepting a point that satisfies only one inequality. Always check a candidate point against every inequality in the system before calling it a solution. A single failure disqualifies the point.
- Assuming the boundary lines' intersection point is always part of the solution set. The intersection point is only a solution if it satisfies both original inequalities, including their symbols. If either inequality is strict at that point, the corner itself is excluded even though it's still the geometric meeting point of the two lines.
- Forgetting that "and" is implied between the inequalities in a system. Even when a problem doesn't explicitly write "and," a system requires all listed inequalities to hold simultaneously.
- Mixing up which inequality failed when a point isn't a solution. Check each inequality separately and keep track of which one the point violates, this often matters for follow-up questions or for choosing a corrected point.
- Overlooking that some systems have no solution at all. If two boundary lines are parallel and their shaded regions face away from each other, there's no overlap and the system's solution set is empty.
Frequently Asked Questions
How is a system of inequalities different from a system of equations?
A system of two linear equations typically has exactly one solution, a single point where the lines cross. A system of two linear inequalities typically has infinitely many solutions, an entire region of the plane, since each inequality itself covers a half-plane rather than a single line.
Can a system of inequalities have more than two inequalities?
Yes. Some systems, especially real-world constraint problems, combine three or more inequalities at once, like x ≥ 0, y ≥ 0, and a budget inequality together. The solution method is the same: a point must satisfy every inequality in the list.
Why do systems of inequalities show up in constraint or "feasible region" word problems?
Real situations often involve more than one limit at once, like a maximum budget and a minimum quantity, and each limit naturally becomes its own inequality. The set of production plans, purchases, or values satisfying every limit at once is exactly the system's solution region.
Where can I practice more problems like these?
The Systems of Inequalities quizzes in the SAT Math Question Bank include 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 on the Digital SAT.