Chapter 2.1, Problem 8QR

To determine

Towrite:The inequality $\left|x-c\right|<d^2$ in the form of $a<x<b$ .

Answer to Problem 8QR

The inequality $\left|x-c\right|<d^2$ in the form of $a<x<b$ is written as $-d^2+c<x<d^2+c$ .

Explanation of Solution

Given information:

The inequality is $\left|x-c\right|<d^2$ .

Calculation:

If a modulus of a variable is less than a positive number $a$ , then the variable lies between $-a$ and $a$ .

  $\begin{array}{c}\left|x\right|<a\\ -a<x<a\end{array}$

Substitute $d^2$ for $a$ and $x-c$ for $x$ in the above inequality.

  $\begin{array}{c}\left|x-c\right|<d^2\\ -d^2<x-c<d^2\end{array}$

Simplify the inequality.

  $\begin{array}{c}-d^2+c<x-c+c<d^2+c\\ -d^2+c<x<d^2+c\end{array}$

Therefore, the inequality $\left|x-c\right|<d^2$ in the form of $a<x<b$ is written as $-d^2+c<x<d^2+c$ .