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

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Solutions of a System of Linear Inequalities

Example

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

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

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

Step 2 — Graph $y < x + 1$ on the same grid. The boundary line is $y = x + 1$ . Because the inequality uses $<$ (strict), draw a dashed line. Test $(0, 0)$ : $0 < 0 + 1$ gives $0 < 1$ , which is true, so shade the side that contains the origin.

Step 3 — Identify the overlapping region. The solution is the area where both shaded regions overlap. The point where the two boundary lines intersect is not included in the solution because it does not satisfy the strict inequality $y < x + 1$ .