Chapter 3.5, Problem 66E

To determine

To write: An inequality which represents the graph.

Answer to Problem 66E

The required inequality is $y>2x-3$ .

Explanation of Solution

Given information:

The graph:

  

Calculation:

From the given graph, it is seen that the line passes through the points $\left(0,-3\right)$ and $\left(1.5,0\right)$ .

Determine the equation of the line using the points $\left(0,-3\right)$ and $\left(1.5,0\right)$ .

For the point $\left(0,-3\right)$ the equation of the line is,

  $\begin{array}{c}y=mx+c\\ -3=m\cdot 0+c\\ c=-3\end{array}$

For the point $\left(1.5,0\right)$ the equation of the line is,

  $\begin{array}{c}y=mx+c\\ 0=m\cdot \left(1.5\right)+c\\ 1.5m+c=0\end{array}$

Substitute $-3$ for c in the above equation.

  $\begin{array}{c}1.5m-3=0\\ \frac{\cancel{1.5}m}{\cancel{1.5}}=\frac{{\cancel{3}}^2}{\cancel{1.5}}\\ m=2\end{array}$

So, the equation of the line is,

  $\begin{array}{c}y=\left(2\right)x+\left(-3\right)\\ y=2x-3\end{array}$

Here, the line $y=2x-3$ divides the graph into two half-spaces.

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

  $\begin{array}{l}0>2\left(0\right)-3\\ 0>-3\end{array}$

Thus, the sign of the inequality is:

  $y>2x-3$

Hence, the required inequality is $y>2x-3$ .