Chapter 5.5, Problem 55PFA

To determine

To calculate:The value of y for the given inequality and given value of x .

Answer to Problem 55PFA

The value of y will be $-7\le y\le 13$ .

Explanation of Solution

Given information:The inequality is $|x+y|\le 5$ and value of xis $-8\le x\le 2$ .

Formula used:

  $\begin{array}{l}\left|x\right|=x,\text{for}x\ge 0\\ \&\\ \left|x\right|=-x,\text{for}x<0\end{array}$

Calculation:

The inequality $|x+y|\le 5$ can be written as

  $x+y\le 5$ and $x+y\ge -5$

If $x+y\le 5$ ,then $y\le 5-x$$\begin{array}{l}-8\le x\le 2\\ \text{mulitply}\text{the}\text{inequality}\text{with}-1,\\ 8\ge -x\ge -2\\ \text{now}\text{add}\text{5}\text{in}\text{the}\text{inequality,}\\ 13\ge 5-x\ge 3\\ 13\ge y\ge 3\end{array}$

If $x+y\ge -5$ ,then $y\ge -5-x$$\begin{array}{l}-8\le x\le 2\\ \text{mulitply}\text{the}\text{inequality}\text{with}-1,\\ 8\ge -x\ge -2\\ \text{now}\text{add}-\text{5}\text{in}\text{the}\text{inequality,}\\ 3\ge -5-x\ge -7\\ 3\ge y\ge -7\end{array}$

Hence the value of y must follow $13\ge y\ge 3$ and $3\ge y\ge -7$ .Therefore the value of y will be

  $-7\le y\le 13$ .