The problem asks us to graph the solution to the following system of inequalities: $y \ge 3x - 1$ $x + y \le 5$

1. Problem Description

The problem asks us to graph the solution to the following system of inequalities:

$y \ge 3x - 1$

$x + y \le 5$

2. Solution Steps

First, we graph the line $y = 3x - 1$. This line has a y-intercept of $-1$ and a slope of $3$. Since the inequality is $y \ge 3x - 1$, we graph the region above the line $y = 3x - 1$. The line itself is included in the solution, so it is a solid line.

Second, we graph the line $x + y = 5$. We can rewrite this as $y = -x + 5$. This line has a y-intercept of $5$ and a slope of $-1$. Since the inequality is $x + y \le 5$, or $y \le -x + 5$, we graph the region below the line $y = -x + 5$. The line itself is included in the solution, so it is a solid line.

The solution to the system of inequalities is the region where the two shaded regions overlap.

3. Final Answer

The solution is the region that satisfies both inequalities. The graph will show two lines: $y = 3x - 1$ and $x+y = 5$ (or $y = -x + 5$). The region above $y = 3x - 1$ and below $x + y = 5$ is the solution. Both lines are solid lines since the inequalities are greater than or equal to and less than or equal to, respectively.