Chapter 3.3, Problem 19E

To determine

the correct ordered pair as a solution for the given inequality and the method to check whether the graph given is correct by using these ordered pairs.

Answer to Problem 19E

  $\left(0,0\right),\left(1,1\right)\text{ and }\left(1,-1\right)$

If $\left(0,0\right),\left(1,1\right)\text{ and }\left(1,-1\right)$ comes under shaded region then the graph is correct.

Explanation of Solution

Given information:

The inequality is shown below.

  $y\le \sqrt{x}+2$

The ordered pairs $\left(x,y\right)$ are $\left(0,0\right),\left(1,4\right),\left(1,1\right),\left(-1,0\right)$ and $\left(1,-1\right)$ .

The graph is shown below.

  

Calculation:

Put the values of $\left(0,0\right)$ in the inequality and simplify as shown below.

  $\begin{array}{l}0\le \sqrt{0}+2\\ 0\le 2\left[Correct\right]\end{array}$

Put the values of $\left(1,4\right)$ in the inequality and simplify as shown below.

  $\begin{array}{l}4\le \sqrt{1}+2\\ 4\le 1+2\\ 4\le 3\left[Incorrect\right]\end{array}$

Put the values of $\left(1,1\right)$ in the inequality and simplify as shown below.

  $\begin{array}{l}1\le \sqrt{1}+2\\ 1\le 1+2\\ 1\le 3\left[Correct\right]\end{array}$

Put the values of $\left(-1,0\right)$ in the inequality and simplify as shown below.

  $\begin{array}{l}0\le \sqrt{-1}+2\\ 0\le undefined+2\left[Incorrect\right]\end{array}$

Put the values of $\left(1,-1\right)$ in the inequality and simplify as shown below.

  $\begin{array}{l}-1\le \sqrt{1}+2\\ -1\le 1+2\\ -1\le 3\left[Correct\right]\end{array}$

Since the ordred pairs $\left(0,0\right),\left(1,1\right)\text{ and }\left(1,-1\right)$ satisfied the given inequality. So, the solution of the given inequality is $\left(0,0\right),\left(1,1\right)\text{ and }\left(1,-1\right)$ .

If $\left(0,0\right),\left(1,1\right)\text{ and }\left(1,-1\right)$ comes under shaded region and

  $\left[\left(1,4\right),\left(-1,0\right)\right]$ in the un-shaded region then the given graph of the inequality is correct otherwise incorrect.