Question

Which description matches the graph of the inequality y ≥ |x + 2| – 3? a shaded region above a solid boundary line a shaded region below a solid boundary line a shaded region below a dashed boundary line a shaded region above a dashed boundary line

Answer 1

The graph of the inequality y ≥ |x + 2| − 3 is represented by a shaded region above a solid V-shaped boundary line that has its vertex at the point (-2, -3).

Step-by-step explanation:

To determine which description matches the graph of the inequality y ≥ |x + 2| − 3, it is essential to understand the properties of the inequality and the properties of absolute value functions. The absolute value function |x + 2| creates a V-shaped graph with a vertex at the point (-2, 0). When this is adjusted by subtracting 3, the vertex of the V shape is moved down to (-2, -3).

The inequality sign ≥ indicates that we are looking for the region that includes all y-values that are greater than or equal to the function |x + 2| − 3. Therefore, the region of the graph that we are interested in is above this V-shaped graph. The equal part of the inequality ≥ allows us to use a solid boundary line rather than a dashed line, because the points on the graph of y = |x + 2| − 3 are included in the solutions for the inequality.

So, the correct description is a shaded region above a solid boundary line, which represents all the points (x, y) where y is greater than or equal to the value of |x + 2| minus 3.

Answer 2

The given inequality is y ≥ |x + 2| -3.

This inequality may be written two ways:
(a) y ≥ x + 2 - 3
or
y ≥ x - 1

(b) y ≥ -x -2 - 3
or
y ≥ -x - 5

A graph of the inequality is shown below. The shaded region satisfies the inequality.

Answer: A shaded region above a solid boundary line.