Question

algebra 1 quiz - systems of linear inequalities
what system of inequalities is represented by the graph?
1.
a. $y \geq x + 3$
$y \leq 2x - 6$
b. $y \geq x - 2$
$y \geq -3x - 6$
c. $y \leq x + 3$
$y \geq 2x - 6$
d. $y \leq x - 2$
$y \leq -3x - 6$
2.
a. $y \leq x + 3$
$8x + y \leq -8$
b. $y \geq x + 8$
$3x + y \leq -8$
c. $y \geq x + 8$
$8x + y \geq -8$
d. $y \leq x + 8$
$3x + y \geq -8$

Explanation:

Step1: Analyze Question 1 lines

First, identify the equations of the two lines in the first graph.

• Line 1 (positive slope): passes through $(3,0)$ and $(0,-2)$, slope $m=\frac{-2-0}{0-3}=\frac{2}{3}$? No, correction: passes through $(2,-2)$ and $(0,-2)$? No, correct: the line with positive slope has intercept $(0,-2)$ and passes through $(3,1)$, so equation $y=x-2$.
• Line 2 (negative slope): passes through $(-2,0)$ and $(0,-6)$, slope $m=\frac{-6-0}{0-(-2)}=-3$, equation $y=-3x-6$.

The shaded region is below both lines, so inequalities are $y \leq x-2$ and $y \leq -3x-6$.

Step2: Match Question1 to options

Compare with given options: this matches option d.

Step3: Analyze Question 2 lines

Identify equations for the second graph:

• Line 1 (positive slope): passes through $(-8,0)$ and $(0,8)$, slope $m=\frac{8-0}{0-(-8)}=1$, equation $y=x+8$.
• Line 2 (negative slope): passes through $(0,-8)$ and $(-\frac{8}{3},0)$, slope $m=\frac{0-(-8)}{-\frac{8}{3}-0}=-3$, equation $3x+y=-8$ or $y=-3x-8$.

The shaded region is above the positive slope line and below the negative slope line, so inequalities are $y \geq x+8$ and $3x+y \leq -8$.

Step4: Match Question2 to options

Compare with given options: this matches option b.

Answer:

1. d. $y \leq x - 2$

$y \leq -3x - 6$

1. b. $y \geq x + 8$

$3x + y \leq -8$