Question

On a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (negative 2, 3) and (0, negative 1). Everything to the left of the line is shaded. The second dashed line has a negative slope and goes through (0, 2) and (1, 0). Everything to the right of the line is shaded.
Which inequality pairs with y≤−2x−1 to complete the system of linear inequalities represented by the graph?

y<−2x+2
y>−2x+2
y<2x−2
y>2x−2

Answer 1

To determine which inequality pairs with \( y \leq -2x - 1 \) in the system of linear inequalities represented by the graph, we first analyze the two lines described:

1. The first solid line with a negative slope goes through the points \((-2, 3)\) and \((0, -1)\). The equation given is \( y = -2x - 1 \), so the inequality associated with this line is \( y \leq -2x - 1 \) which means everything to the left of this line is shaded.

2. The second dashed line goes through the points \((0, 2)\) and \((1, 0)\) which has a negative slope. To find its equation, we calculate the slope:
\[
\text{slope} = \frac{0 - 2}{1 - 0} = -2
\]
Therefore, the equation of the line can be expressed as \( y = -2x + 2 \).

Since the line is dashed, the inequality associated with this line does not include equality, which means it will be either \( y < -2x + 2 \) or \( y > -2x + 2 \).

Given that everything to the right of this line is shaded, the correct inequality must represent the region to the right of the line. Thus, it must be:

\[
y < -2x + 2
\]

So, the correct choice to complete the system of linear inequalities represented by the graph is:

\[
\boxed{y < -2x + 2}
\]