Question

solve the system of inequalities by graphing.
$x \leq 3$
$y < 4x - 9$
select a line to change it between solid and dotted. select a region to shade it.

Explanation:

Step1: Analyze \( x \leq 3 \)

The inequality \( x \leq 3 \) is a vertical line at \( x = 3 \). Since the inequality is "less than or equal to", the line should be solid. The region to shade is to the left of \( x = 3 \) (all \( x \)-values less than or equal to 3).

Step2: Analyze \( y < 4x - 9 \)

The inequality \( y < 4x - 9 \) is a linear inequality. First, graph the line \( y = 4x - 9 \). Since the inequality is "less than" (not "less than or equal to"), the line should be dotted. To find the region to shade, test a point not on the line, e.g., \( (0,0) \): \( 0 < 4(0) - 9 \) simplifies to \( 0 < -9 \), which is false. So we shade the region below the dotted line \( y = 4x - 9 \) (where the inequality holds).

Step3: Find the Intersection

The solution to the system is the region where both inequalities are satisfied: left of the solid line \( x = 3 \) and below the dotted line \( y = 4x - 9 \).

Answer:

To solve the system:

1. Graph \( x \leq 3 \) as a solid vertical line at \( x = 3 \), shading left of the line.
2. Graph \( y < 4x - 9 \) as a dotted line (using \( y = 4x - 9 \)), shading below the line.
3. The solution is the overlapping region (left of \( x = 3 \) and below \( y = 4x - 9 \)).

(Note: On the provided graph, set \( x \leq 3 \) to solid, \( y < 4x - 9 \) to dotted, and shade the intersection.)