1Cademy - Solving $$\{x + y \leq 2,\; y \geq \frac{2}{3}x

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Solving a System of Linear Inequalities by Graphing

Example

Solving $\{x + y \leq 2,; y \geq \frac{2}{3}x - 1\}$ by Graphing

Solve the system $\left\{\begin{array}{l} x + y \leq 2 y \geq \frac{2}{3}x - 1 \end{array}\right.$ by graphing.

Step 1 — Graph $x + y \leq 2$ . The boundary line is $x + y = 2$ . Because the inequality uses $\leq$ (non-strict), draw a solid line. Test (0, 0): $0 + 0 \leq 2$ gives $0 \leq 2$ , which is true, so shade the side that contains the origin.

Step 2 — Graph $y \geq \frac{2}{3}x - 1$ on the same grid. The boundary line is $y = \frac{2}{3}x - 1$ . Because the inequality uses $\geq$ (non-strict), draw a solid line. Test (0, 0): $0 \geq \frac{2}{3}(0) - 1$ gives $0 \geq -1$ , which is true, so shade the side that contains the origin.

Step 3 — Identify the solution. The solution is the overlapping shaded region. The intersection point of the boundary lines is included in the solution because both boundary lines are solid.

Step 4 — Verify with a test point. Choose a test point in the overlapping region and substitute it into both inequalities to confirm both are true.

Updated 2026-06-03

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OpenStax Intermediate Algebra

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