Graph the inequality y ≥ -2x – 6
The Correct Answer and Explanation is:
To graph the inequality y ≥ -2x – 6, begin by considering the related equation y = -2x – 6, which represents a straight line in slope-intercept form, where the slope is -2 and the y-intercept is -6.
Plot the line by first locating the y-intercept at the point (0, -6) on the coordinate plane. From that point, use the slope of -2, which indicates a downward direction: for every 1 unit increase in x, y decreases by 2 units. Move right 1 unit and down 2 units to find a second point. Connect the points to form a solid line because the inequality includes “equal to” (≥), which means the points on the line are part of the solution set.
Next, determine which side of the line to shade. Since the inequality is y ≥ -2x – 6, the region above the line should be shaded. This region includes all the points for which the y-values are greater than or equal to the corresponding values on the line. A simple method to verify this involves testing a point not on the line, such as (0, 0). Substituting into the inequality gives:
0 ≥ -2(0) – 6
0 ≥ -6 — which is true.
Since the point (0, 0) satisfies the inequality, the region containing this point should be shaded.
The final graph consists of a solid diagonal line descending from left to right, intersecting the y-axis at -6, and shading the region above this line, representing all coordinate pairs (x, y) that satisfy the inequality. This shaded area indicates the set of solutions to the inequality y ≥ -2x – 6.
