Chapter 3.5, Problem 65E

To determine

To write: An inequality which represents the graph.

Answer to Problem 65E

The required inequality is $y\ge -x+1$ .

Explanation of Solution

Given information:

The graph:

  

Calculation:

From the given graph, it is seen that the line passes through the points $\left(0,1\right)$ and $\left(1,0\right)$ .

Determine the equation of the line using the points $\left(0,1\right)$ and $\left(1,0\right)$ .

For the point $\left(1,0\right)$ the equation of the line is,

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

For the point $\left(0,1\right)$ the equation of the line is,

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

Substitute 1 for c in the above equation.

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

So, the equation of the line is,

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

Here, the line $y=-x+1$ divides the graph into two half-spaces.

Take an arbitrary point that belongs to the solution to determine the sign of the required inequality.

  $\begin{array}{l}2\ge -1+1\\ 2\ge 0\end{array}$

Thus, the sign of the inequality is:

  $y\ge -x+1$

Hence, the required inequality is $y\ge -x+1$ .