Question

graph this inequality:
x ≤ 3
plot points on the boundary line. select the line to switch between solid and dotted. select a
region to shade it.

Explanation:

Step1: Identify the boundary line

The inequality is \( x \leq 3 \). This is a vertical line (since it has the form \( x = \text{constant} \)) at \( x = 3 \). Since the inequality is \( \leq \), the boundary line should be solid (because the points on the line are included in the solution set).

Step2: Plot the boundary line

On the coordinate grid, find the vertical line where \( x = 3 \) (this is a vertical line passing through the point \( (3, y) \) for all \( y \)-values). Draw this line as a solid line.

Step3: Determine the region to shade

To find which region to shade, we can test a point not on the line. A common test point is the origin \( (0, 0) \). Substitute \( x = 0 \) into the inequality: \( 0 \leq 3 \), which is true. So the region containing the origin (all points where \( x \leq 3 \)) should be shaded. This region is to the left of the vertical line \( x = 3 \).

(Note: Since the problem asks to graph it, the key steps are drawing the solid vertical line at \( x = 3 \) and shading the region to the left of this line.)

Answer:

1. Draw a solid vertical line at \( x = 3 \) (since \( x \leq 3 \) includes the line \( x = 3 \)).
2. Shade the region to the left of the line \( x = 3 \) (this includes all points where \( x \)-coordinate is less than or equal to 3).