Question

graph the solution to the following system of inequalities.
y<4x+5
y≥-3x-4

Explanation:

Step1: Graph \( y < 4x + 5 \)

First, graph the line \( y = 4x + 5 \). The slope \( m = 4 \) and the y - intercept \( b = 5 \). Since the inequality is \( y < 4x + 5 \), the line should be dashed (because the inequality is strict, \( y\) is not equal to \( 4x + 5 \)). To determine the region to shade, pick a test point not on the line, e.g., \( (0,0) \). Substitute into the inequality: \( 0<4(0)+5\Rightarrow0 < 5 \), which is true. So shade the region below the dashed line \( y = 4x + 5 \).

Step2: Graph \( y\geq - 3x - 4 \)

Graph the line \( y=-3x - 4 \). The slope \( m=-3 \) and the y - intercept \( b = - 4 \). Since the inequality is \( y\geq - 3x - 4 \), the line should be solid (because \( y\) can be equal to \( - 3x - 4 \)). Pick a test point not on the line, e.g., \( (0,0) \). Substitute into the inequality: \( 0\geq-3(0)-4\Rightarrow0\geq - 4 \), which is true. So shade the region above the solid line \( y=-3x - 4 \).

Step3: Find the Intersection Region

The solution to the system of inequalities is the region that is shaded for both inequalities. So it is the region that is below the dashed line \( y = 4x + 5 \) and above the solid line \( y=-3x - 4 \).

(Note: Since the problem asks to graph the solution, the final answer is the graphical region described above. If we were to describe the steps for graphing, the key elements are the dashed line \( y = 4x+5 \) with shading below, the solid line \( y=-3x - 4 \) with shading above, and the overlapping region as the solution.)

Answer:

The solution is the region below the dashed line \( y = 4x + 5 \) and above the solid line \( y=-3x - 4 \) (graphically, this is the area where the shadings from both inequalities overlap). To graph it:

1. Draw a dashed line for \( y = 4x + 5 \) (slope 4, y - intercept 5) and shade below it.
2. Draw a solid line for \( y=-3x - 4 \) (slope - 3, y - intercept - 4) and shade above it.
3. The overlapping shaded region is the solution.