Question

graph this inequality:
x ≤ 1
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 \( x \leq 1 \). The boundary line is \( x = 1 \), which is a vertical line. Since the inequality is "less than or equal to", the line should be solid (because the points on the line are included in the solution set).

Step2: Plot the boundary line

To plot \( x = 1 \), we can choose any \( y \)-values. For example, when \( y = 0 \), the point is \( (1, 0) \); when \( y = 2 \), the point is \( (1, 2) \); when \( y = -3 \), the point is \( (1, -3) \). Plot these points (or any other points with \( x = 1 \)) on the graph.

Step3: Determine the region to shade

We need to shade the region where \( x \leq 1 \). This is the region to the left of the line \( x = 1 \) (since all points with \( x \)-coordinate less than or equal to 1 satisfy the inequality).

To graph \( x \leq 1 \):

1. Draw a solid vertical line at \( x = 1 \) (plot points like \( (1, 0) \), \( (1, 2) \), \( (1, -3) \) on this line).
2. Shade the region to the left of the line \( x = 1 \) (all points where \( x \)-value is less than or equal to 1).

(Note: Since this is a text-based explanation, the actual graphing would be done on the provided coordinate plane by plotting the boundary line points, setting the line to solid, and shading the left region.)

Answer:

Step1: Identify the boundary line

The inequality is \( x \leq 1 \). The boundary line is \( x = 1 \), which is a vertical line. Since the inequality is "less than or equal to", the line should be solid (because the points on the line are included in the solution set).

Step2: Plot the boundary line

To plot \( x = 1 \), we can choose any \( y \)-values. For example, when \( y = 0 \), the point is \( (1, 0) \); when \( y = 2 \), the point is \( (1, 2) \); when \( y = -3 \), the point is \( (1, -3) \). Plot these points (or any other points with \( x = 1 \)) on the graph.

Step3: Determine the region to shade

We need to shade the region where \( x \leq 1 \). This is the region to the left of the line \( x = 1 \) (since all points with \( x \)-coordinate less than or equal to 1 satisfy the inequality).

To graph \( x \leq 1 \):

1. Draw a solid vertical line at \( x = 1 \) (plot points like \( (1, 0) \), \( (1, 2) \), \( (1, -3) \) on this line).
2. Shade the region to the left of the line \( x = 1 \) (all points where \( x \)-value is less than or equal to 1).

(Note: Since this is a text-based explanation, the actual graphing would be done on the provided coordinate plane by plotting the boundary line points, setting the line to solid, and shading the left region.)