Question

graph the solution of the system of linear inequalities
\\(\

$$\begin{cases}y\\geq x + 2\\y\\geq4 - x\\end{cases}$$

\\) use the graphing tool to graph the inequality. click to enlarge graph

Explanation:

Step1: Identify the boundary - lines

For $y\geq x + 2$, the boundary - line is $y=x + 2$; for $y\geq4 - x$, the boundary - line is $y=4 - x$.

Step2: Graph the boundary - line $y=x + 2$

The slope is $m = 1$ and the y - intercept is $b = 2$. Plot the y - intercept $(0,2)$ and use the slope to find another point. Since the inequality is $\geq$, the line is solid.

Step3: Graph the boundary - line $y=4 - x$

The slope is $m=-1$ and the y - intercept is $b = 4$. Plot the y - intercept $(0,4)$ and use the slope to find another point. Since the inequality is $\geq$, the line is solid.

Step4: Test a point for each inequality

For $y\geq x + 2$, test the point $(0,0)$. Substitute $x = 0$ and $y = 0$: $0\geq0 + 2$ is false, so the solution region for $y\geq x + 2$ is above the line $y=x + 2$.
For $y\geq4 - x$, test the point $(0,0)$. Substitute $x = 0$ and $y = 0$: $0\geq4-0$ is false, so the solution region for $y\geq4 - x$ is above the line $y=4 - x$.

Step5: Find the intersection of the solution regions

The intersection of the regions above $y=x + 2$ and above $y=4 - x$ is the solution of the system.

Answer:

Graph the solid line $y=x + 2$ (slope 1, y - intercept 2) and shade the region above it. Graph the solid line $y=4 - x$ (slope - 1, y - intercept 4) and shade the region above it. The intersection of the two shaded regions is the solution of the system of linear inequalities.