1Cademy - Solving $$\{x - y > 3,\; y < -\frac{1}{5}x + 4\}$$ by Graphing

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Solutions of a System of Linear Inequalities

Example

Solving $\{x - y > 3,\; y < -\frac{1}{5}x + 4\}$ by Graphing

Solve the system $\left\{\begin{array}{l} x - y > 3 \\ y < -\frac{1}{5}x + 4 \end{array}\right.$ by graphing.

Graph $x - y > 3$ . The boundary line is $x - y = 3$ , which has intercepts $x = 3$ and $y = -3$ . Because the inequality uses $>$ (strict), draw a dashed line. Test $(0, 0)$ : $0 - 0 > 3$ gives $0 > 3$ , which is false. So shade the side that does not contain the origin.

Graph $y < -\frac{1}{5}x + 4$ on the same grid. The boundary line is $y = -\frac{1}{5}x + 4$ , which has slope $m = -\frac{1}{5}$ and y-intercept $b = 4$ . Because the inequality uses $<$ (strict), draw a dashed line. Test $(0, 0)$ : $0 < -\frac{1}{5}(0) + 4$ gives $0 < 4$ , which is true. So shade the side that contains the origin.

Identify the solution. The solution is the darker-shaded region where both shadings overlap. The intersection point of the two boundary lines is not included because both lines are dashed.

Verify by choosing a test point in the overlapping region and substituting it into both inequalities to confirm both are true.

Updated 2026-04-28

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