Chapter 3.8, Problem 5P

(a)

To determine

The inequality as either union or the intersection of two sets is to be solved.

Answer to Problem 5P

The inequality as the intersection of two sets is,

  $\left\{x|x\ge 3\right\}\cap \left\{x|x<6\right\}$

Explanation of Solution

The given inequality is,

  $8\le x+5<11$

The given inequality is a compound inequality.

Subtract $5$ to each expression

  $8-5\le x+5-5<11-5$

  $3\le x<6$

The solutions of the inequality are given by

  $3\le x<6$

The solutions of the inequality can be written as follows:

  $x\ge 3\text{and}x<6$

The given inequality is the intersections of two sets which is written as follows:

  $\left\{x|x\ge 3\right\}\cap \left\{x|x<6\right\}$

(b)

To determine

The inequality as either union or the intersection of two sets is to be solved.

Answer to Problem 5P

The inequality as the union of two sets is,

  $\left\{x|x<-2\right\}\cup \left\{x|x>5\right\}$

Explanation of Solution

The given inequality is,

  $\left|4x-6\right|>14$

The given inequality represents the absolute value equation.

It can be written as:

  $4x-6>14$ ................... (1)

  $4x-6<\text{−}14$.......... (2)

solve equation (1)

  $\begin{array}{l}4x-6+6>14+6\\ 4x>20\\ x>5\end{array}$

Similarly equation (2) is solved.

  $\begin{array}{l}4x-6+6<-14+6\\ 4x<-8\\ x<-2\end{array}$

The solutions of the inequality can be written as follows:

  $x<-2\text{and}x>5$

The solution of the given inequality represents disjoints sets.

The compound inequality is the union of two sets which can be written as follows:

  $\left\{x|x<-2\right\}\cup \left\{x|x>5\right\}$