1Cademy - Solving $$\left\{y > \frac{1}{2}x - 4,\; x

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Solving $\left\{y > \frac{1}{2}x - 4,\; x - 2y < -4\right\}$ by Graphing

To solve the system of linear inequalities $\left\{\begin{array}{l} y > \frac{1}{2}x - 4 \\ x - 2y < -4 \end{array}\right.$ by graphing, follow these steps:

Step 1: Graph the first inequality, $y > \frac{1}{2}x - 4$ . The boundary line is $y = \frac{1}{2}x - 4$ , which can be graphed using its slope ( $m = \frac{1}{2}$ ) and y-intercept ( $b = -4$ ). The boundary line must be dashed because the inequality uses the strict $>$ symbol. Testing the origin $(0, 0)$ yields a true statement, so shade the region that contains $(0, 0)$ .

Step 2: Graph the second inequality, $x - 2y < -4$ , on the same coordinate plane. Its boundary line, $x - 2y = -4$ , has intercepts at $x = -4$ and $y = 2$ . This boundary line is also dashed due to the strict $<$ symbol. Testing the origin $(0, 0)$ yields a false statement, so shade the region that does not contain $(0, 0)$ .

Step 3: Identify the solution. No points on either boundary line are included in the solution since both lines are dashed. The solution is the overlapping area where both regions are shaded twice. In this specific case, the doubly-shaded region exactly matches the solution set for the single inequality $x - 2y < -4$ .

Updated 2026-05-12

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