Question

identifying the solution to a system of two - variable inequalities
which equation represents an inequality in the system of inequalities shown in the graph?
y > - 2x - 1
which point is a solution to the system?
(-4, -2)
(-1, -2)
(1, 4)
(2, 2)

Explanation:

Step1: Identify the second inequality

First, find the equation of the solid line. It passes through (0, -1) and (1, 1). The slope $m=\frac{1-(-1)}{1-0}=2$. Using slope-intercept form $y=mx+b$, $b=-1$, so the line is $y=2x-1$. The shaded region is above this solid line, so the inequality is $y\geq2x-1$.

Step2: Test point (-4, -2)

Substitute into $y>-2x-1$: $-2 > -2(-4)-1
ightarrow -2 > 8-1
ightarrow -2>7$, which is false.

Step3: Test point (-1, -2)

Substitute into $y>-2x-1$: $-2 > -2(-1)-1
ightarrow -2 > 2-1
ightarrow -2>1$, which is false.

Step4: Test point (1, 4)

Substitute into $y>-2x-1$: $4 > -2(1)-1
ightarrow 4 > -3$, which is true. Substitute into $y\geq2x-1$: $4\geq2(1)-1
ightarrow 4\geq1$, which is true.

Step5: Test point (2, 2)

Substitute into $y\geq2x-1$: $2\geq2(2)-1
ightarrow 2\geq3$, which is false.

Answer:

(1, 4)