Chapter 3.1, Problem 55PPSE

Solution

To identify which inequality represents the graph given in the problem.

The given graph is the solution set of $y>\frac{1}{2}\left|x\right|+1$ .

Given the graph:

  

Calculation:

Observe that the red dotted line represented in the graph has the equation $y=\frac{1}{2}\left|x\right|+1$ .

Now, since the line is not included in the solution set represented in the graph, the answer is either $y<\frac{1}{2}\left|x\right|+1$ or $y>\frac{1}{2}\left|x\right|+1$ .

Observe that the point $\left(0,2\right)$ lies in the solution set represented in the graph.

Substitute $x=0\text{and }y=2$ in the inequality $y<\frac{1}{2}\left|x\right|+1$ :

  $2<\frac{1}{2}\left|0\right|+1$ .

Simplify the inequality $2<\frac{1}{2}\left|0\right|+1$ to check whether it is valid:

  $\begin{array}{l}2<\frac{1}{2}\left|0\right|+1\\ 2<\frac{1}{2}\cdot 0+1\\ 2<0+1\\ 2<1\end{array}$ .

Observe that $2<1$ is an invalid inequality.

Thus, $\left(0,2\right)$ does not lie in the solution set of $y<\frac{1}{2}\left|x\right|+1$ .

That is, the given graph is not the solution set of $y<\frac{1}{2}\left|x\right|+1$ .

Then, it follows that, the given graph is the solution set of $y>\frac{1}{2}\left|x\right|+1$ .

Conclusion:

The given graph is the solution set of $y>\frac{1}{2}\left|x\right|+1$ .