Key takeaway: A system of inequalities is a set of two or more inequalities in the same variables, and its solution is the region where all of them are true at once.
Grade 9 – Algebra 1A-REI.D.12Expressions and Equations
Definition of System of Inequalities
A system of inequalities asks for points that satisfy every inequality in the list at the same time. Each inequality on its own shades half of the coordinate plane. When you graph all of them together, the region where every shading overlaps is the solution set.
Boundary lines come from replacing the inequality sign with an equals sign. Use a solid line for \(\le\) or \(\ge\), where points on the line count, and a dashed line for \(<\) or \(>\), where they do not. Systems model limits that must all hold, for example a budget, a time limit and a minimum order.
Formula and notation
The inequalities are listed together, usually one under another or joined with “and”. For example:
\[y \le -x + 6 \quad\text{and}\quad y > 2x – 3\]
A point \((x, y)\) is a solution only if it makes both statements true.
Key facts
- The answer to a system of inequalities is an overlapping region, not one point.
- Dashed boundary lines mean points on the line are excluded.
- Test one point not on a boundary line to decide which side to shade.
- Some systems have no overlap, so the solution set is empty.
System of Inequalities example
Is the point \((1, 2)\) a solution of the system \(y \le -x + 6\) and \(y > 2x – 3\)? What about \((4, 1)\)?
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Test \((1, 2)\) in the first: \(2 \le -1 + 6 = 5\), true. In the second: \(2 > 2(1) – 3 = -1\), true.
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Test \((4, 1)\) in the first: \(1 \le -4 + 6 = 2\), true.
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In the second: \(1 > 2(4) – 3 = 5\) is false, so \((4, 1)\) fails.
\((1, 2)\) is a solution; \((4, 1)\) is not, since it breaks the second inequality.
Common mistake
Watch out
Sol checks \((4, 1)\) only in the first inequality, sees that it is true, and declares the point a solution.
Why it is wrong
A solution must satisfy every inequality, so test each one. The point \((4, 1)\) passes \(y \le -x + 6\) but fails \(y > 2x – 3\), which means it lies outside the shaded overlap. One false statement is enough to reject a point.
Related terms and lessons
- Inequality
- Compound Inequality
- System of Equations
- Linear Equation
- Coordinate Plane
- Slope-Intercept Form
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