Chapter M, Problem 1DE

Graph:

$\begin{array}{l}x+y\ge 1,\\ y-x\ge 2.\end{array}$

To determine

To graph: The system of inequalities $x+y\ge 1$ and $y-x\ge 2$

Explanation of Solution

Given Information:

The system of inequalities $x+y\ge 1$ and $y-x\ge 2$.

Graph:

Consider the provided system of inequalities.

To graph the linear inequality $x+y\ge 1$, first graph the equation $x+y=1$.

Compute some points on the graph of linear equation $x+y=1$.

Substitute $x=0$,

$\begin{array}{c}0+y=1\\ y=1\end{array}$

So, $\left(0,1\right)$ is a point on the graph of $x+y=1$.

Substitute $y=0$,

$\begin{array}{c}x+0=1\\ x=1\end{array}$

So, $\left(1,0\right)$ is a point on the graph of $x+y=1$.

Substitute $x=-1$,

$\begin{array}{c}-1+y=1\\ y=1+1\\ =2\end{array}$

So, $\left(-1,2\right)$ is a point on the graph of $x+y=1$.

Plot all three points $\left(0,1\right)$, $\left(1,0\right)$ and $\left(-1,2\right)$ on a graph paper and join them with a solid line.

The graph of the linear equation $x+y=1$ is as follows:

Take $\left(0,0\right)$ as the test point and check it in the inequality $x+y\ge 1$,

$\begin{array}{c}x+y\ge 1\\ 0+0\ge 1\\ 0\ge 1\end{array}$

This is false.

So, the region on the opposite side of point $\left(0,0\right)$ is the solution of the inequality $x+y\ge 1$.

Shade all points on that side of the line using pink shading.

To graph the linear inequality $y-x\ge 2$, first graph the equation $y-x=2$.

Compute some points on the graph of linear equation $y-x=2$.

Substitute $x=0$,

$\begin{array}{c}y-0=2\\ y=2\end{array}$

So, $\left(0,2\right)$ is a point on the graph of $y-x=2$.

Substitute $y=0$,

$\begin{array}{c}0-x=2\\ -x=2\\ x=-2\end{array}$

So, $\left(-2,0\right)$ is a point on the graph of $y-x=2$.

Substitute $x=-1$,

$\begin{array}{c}y-\left(-1\right)=2\\ y=2+\left(-1\right)\\ =1\end{array}$

So, $\left(-1,1\right)$ is a point on the graph of $y-x=2$.

Plot all three points $\left(0,2\right)$, $\left(-2,0\right)$ and $\left(-1,1\right)$ on a graph paper and join them with a solid line.

The graph of the linear equation $y-x=2$ is as follows:

Take $\left(0,0\right)$ as the test point and check it in the inequality $y-x\ge 2$,

$\begin{array}{c}y-x\ge 2\\ 0-0\ge 2\\ 0\ge 2\end{array}$

This is false.

So, the region on the opposite side of point $\left(0,0\right)$ is the solution of the inequality $y-x\ge 2$.

Shade all points on that side of the line using sky blue shading.

Draw both inequalities on the same graph paper. The intersection of two graphs is as follows

The grey region in the above graph represents the solution of the system of inequalities.