Question

graph the intersection of $x \geq 1$ and $y \leq -8$. use the graphing tool to graph the intersection. click to enlarge graph

Explanation:

Step1: Analyze \( x \geq 1 \)

The inequality \( x \geq 1 \) represents all the points to the right of and including the vertical line \( x = 1 \). To graph this, draw a vertical line at \( x = 1 \) (solid line because the inequality is "greater than or equal to") and shade the region where \( x \) values are 1 or larger (to the right of the line).

Step2: Analyze \( y \leq -8 \)

The inequality \( y \leq -8 \) represents all the points below and including the horizontal line \( y = -8 \). To graph this, draw a horizontal line at \( y = -8 \) (solid line because the inequality is "less than or equal to") and shade the region where \( y \) values are -8 or smaller (below the line).

Step3: Find the Intersection

The intersection of \( x \geq 1 \) and \( y \leq -8 \) is the region that is both to the right of (or on) \( x = 1 \) and below (or on) \( y = -8 \). So, the graph will be the area where these two shaded regions overlap. This region is a quadrant - like area bounded on the left by \( x = 1 \) (vertical solid line) and below by \( y=-8\) (horizontal solid line), and it extends infinitely to the right and downwards (since there are no upper or right - hand bounds other than the natural extension of the inequalities).

Graph Description (for the graphing tool):

1. Draw a vertical solid line at \( x = 1 \). Shade the region to the right of this line (including the line itself).
2. Draw a horizontal solid line at \( y=-8\). Shade the region below this line (including the line itself).
3. The intersection (the final graph) is the region that is shaded in both of the above shadings, i.e., the area that is to the right of \( x = 1 \) and below \( y=-8\).

(Note: Since the problem asks to use the graphing tool, the above steps guide how to create the graph. If we were to describe the graph's appearance, it's a region in the fourth - like quadrant relative to the point \( (1, - 8) \), with the left - hand boundary \( x = 1 \) (vertical) and lower boundary \( y=-8\) (horizontal), and extending infinitely in the positive \( x \) - direction and negative \( y \) - direction.)

Answer:

Step1: Analyze \( x \geq 1 \)

The inequality \( x \geq 1 \) represents all the points to the right of and including the vertical line \( x = 1 \). To graph this, draw a vertical line at \( x = 1 \) (solid line because the inequality is "greater than or equal to") and shade the region where \( x \) values are 1 or larger (to the right of the line).

Step2: Analyze \( y \leq -8 \)

The inequality \( y \leq -8 \) represents all the points below and including the horizontal line \( y = -8 \). To graph this, draw a horizontal line at \( y = -8 \) (solid line because the inequality is "less than or equal to") and shade the region where \( y \) values are -8 or smaller (below the line).

Step3: Find the Intersection

The intersection of \( x \geq 1 \) and \( y \leq -8 \) is the region that is both to the right of (or on) \( x = 1 \) and below (or on) \( y = -8 \). So, the graph will be the area where these two shaded regions overlap. This region is a quadrant - like area bounded on the left by \( x = 1 \) (vertical solid line) and below by \( y=-8\) (horizontal solid line), and it extends infinitely to the right and downwards (since there are no upper or right - hand bounds other than the natural extension of the inequalities).

Graph Description (for the graphing tool):

1. Draw a vertical solid line at \( x = 1 \). Shade the region to the right of this line (including the line itself).
2. Draw a horizontal solid line at \( y=-8\). Shade the region below this line (including the line itself).
3. The intersection (the final graph) is the region that is shaded in both of the above shadings, i.e., the area that is to the right of \( x = 1 \) and below \( y=-8\).

(Note: Since the problem asks to use the graphing tool, the above steps guide how to create the graph. If we were to describe the graph's appearance, it's a region in the fourth - like quadrant relative to the point \( (1, - 8) \), with the left - hand boundary \( x = 1 \) (vertical) and lower boundary \( y=-8\) (horizontal), and extending infinitely in the positive \( x \) - direction and negative \( y \) - direction.)