1Cademy - Solving $$\left\{y \geq 3x - 2,\; y

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Solving a System of Linear Inequalities by Graphing

Example

Solving $\left\{y \geq 3x - 2,\; y < -1\right\}$ by Graphing

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

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

Step 2 — Graph $y < -1$ on the same grid. The boundary line is $y = -1$ , a horizontal line passing through $y = -1$ . Because the inequality uses $<$ (strict), draw a dashed line. Test $(0, 0)$ : $0 < -1$ is false, so shade the side that does not contain the origin.

Step 3 — Identify the solution. The solution is the area where both shaded regions overlap. The intersection point of the boundary lines is not included because one of the boundary lines is dashed.

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 2026-04-28

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OpenStax Intermediate Algebra

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