Question

which ordered pairs are in the solution set of the system of linear inequalities?
$y \geq -\frac{1}{2}x$
$y < \frac{1}{2}x + 1$
(5, -2), (3, 1), (-4, 2)
(5, -2), (3, -1), (-4, -3)
(5, -2), (3, 1), (4, 2)
(5, -2), (-3, 1), (4, 2)

Explanation:

Step1: Test first inequality $y\geq -\frac{1}{2}x$

For each ordered pair $(x,y)$, substitute into the inequality:

• $(5,-2)$: $-2\geq -\frac{1}{2}(5) \implies -2\geq -2.5$ ✔️
• $(3,1)$: $1\geq -\frac{1}{2}(3) \implies 1\geq -1.5$ ✔️
• $(-4,2)$: $2\geq -\frac{1}{2}(-4) \implies 2\geq 2$ ✔️
• $(3,-1)$: $-1\geq -\frac{1}{2}(3) \implies -1\geq -1.5$ ✔️
• $(4,-3)$: $-3\geq -\frac{1}{2}(4) \implies -3\geq -2$ ❌
• $(4,2)$: $2\geq -\frac{1}{2}(4) \implies 2\geq -2$ ✔️
• $(-3,1)$: $1\geq -\frac{1}{2}(-3) \implies 1\geq 1.5$ ❌

Step2: Test second inequality $y<\frac{1}{2}x+1$

Test the remaining valid pairs from Step1:

• $(5,-2)$: $-2<\frac{1}{2}(5)+1 \implies -2<3.5$ ✔️
• $(3,1)$: $1<\frac{1}{2}(3)+1 \implies 1<2.5$ ✔️
• $(-4,2)$: $2<\frac{1}{2}(-4)+1 \implies 2<-1$ ❌
• $(3,-1)$: $-1<\frac{1}{2}(3)+1 \implies -1<2.5$ ✔️
• $(4,2)$: $2<\frac{1}{2}(4)+1 \implies 2<3$ ✔️

Answer:

○ (5, -2), (3, 1), (3, -1), (4, 2)