The problem requires graphing the solution to the following system of inequalities: $y \ge 4x - 3$ $x + y \le 7$

1. Problem Description

The problem requires graphing the solution to the following system of inequalities:

$y \ge 4x - 3$

$x + y \le 7$

2. Solution Steps

First, we need to graph the line $y = 4x - 3$. This is a line with slope $4$ and y-intercept $-3$. We can find two points on the line to plot.

If $x = 0$, then $y = 4(0) - 3 = -3$. So, $(0, -3)$ is on the line.

If $x = 1$, then $y = 4(1) - 3 = 1$. So, $(1, 1)$ is on the line.

Since $y \ge 4x - 3$, we shade the region above the line $y = 4x - 3$.

Second, we need to graph the line $x + y = 7$. We can rewrite this as $y = -x + 7$. This is a line with slope $-1$ and y-intercept $7$. We can find two points on the line to plot.

If $x = 0$, then $y = -0 + 7 = 7$. So, $(0, 7)$ is on the line.

If $x = 7$, then $y = -7 + 7 = 0$. So, $(7, 0)$ is on the line.

Since $x + y \le 7$, or $y \le -x + 7$, we shade the region below the line $y = -x + 7$.

The solution to the system of inequalities is the region where the two shaded regions overlap. The line $y = 4x - 3$ is solid because the inequality is $\ge$. The line $x + y = 7$ is solid because the inequality is $\le$.

3. Final Answer

The solution is the region where $y \ge 4x - 3$ and $x + y \le 7$. This is found by graphing the two inequalities and identifying the area of overlap.