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

Answer: The first line with a negative slope passing through (-2, 3) and (0, -1) can be represented as:

y - (-1) = (3 - (-1)) / (-2 - 0) * (x - 0)

y + 1 = -2/2 * x

y = -2x - 1

The second line with a negative slope passing through (0, 2) and (1, 0) can be represented as:

y - 0 = (2 - 0) / (0 - 1) * (x - 1)

y = -2x + 2

Since everything to the left of the first line is shaded and everything to the right of the second line is shaded, we want to find an inequality that satisfies both conditions, i.e., an inequality that is true for all points that are shaded.

From the graph, we see that the shaded region is below the first line and to the right of the second line. Therefore, the inequality that completes the system of linear inequalities is:

y ≤ -2x - 1 AND y > -2x + 2

Simplifying this system of inequalities, we get:

y > -2x + 2 - -2x - 1

y > 3

Therefore, the inequality that pairs with y ≤ -2x - 1 to complete the system of linear inequalities represented by the graph is:

y > 3.

Explanation: