We are asked to graph the solution to the following system of inequalities: $y \ge 2x - 1$ $x + y \le 3$

1. Problem Description

We are asked to graph the solution to the following system of inequalities:

$y \ge 2x - 1$

$x + y \le 3$

2. Solution Steps

First, we will graph the line $y = 2x - 1$. This is a line with slope $2$ and y-intercept $-1$.

When $x=0$, $y = 2(0) - 1 = -1$. So the point $(0, -1)$ is on the line.

When $x=1$, $y = 2(1) - 1 = 1$. So the point $(1, 1)$ is on the line.

Since the inequality is $y \ge 2x - 1$, we shade the region above the line. Because the inequality is $\ge$, we draw the line as a solid line.

Second, we will graph the line $x + y = 3$. We can rewrite this as $y = -x + 3$. This is a line with slope $-1$ and y-intercept $3$.

When $x=0$, $y = -0 + 3 = 3$. So the point $(0, 3)$ is on the line.

When $x=3$, $y = -3 + 3 = 0$. So the point $(3, 0)$ is on the line.

Since the inequality is $x + y \le 3$, we shade the region below the line. Because the inequality is $\le$, we draw the line as a solid line.

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

3. Final Answer

The solution is the intersection of the regions $y \ge 2x - 1$ and $x + y \le 3$. To graph it using the graphing tool, one would:

1. Graph the line $y = 2x - 1$ as a solid line, and shade above the line.

2. Graph the line $x + y = 3$ (or $y = -x + 3$) as a solid line, and shade below the line.

The overlapping shaded region represents the solution set to the system of inequalities.