Chapter 1.1, Problem 73E

a)

To determine

To find: the distance between the two numbers $2$ and $17$ .

Answer to Problem 73E

The distance between the two numbers is $15$ .

Explanation of Solution

Given information:

Two numbers are given as

  $2$ and $17$

Formula used:

Distance between two numbers a and b is given by

  $d\left(a,b\right)=\left|a-b\right|=\left|b-a\right|=d\left(b,a\right)$

Absolute value property:

For any real number a,

  $\begin{array}{l}\left|a\right|=a\ge 0\\ \left|-a\right|=\left|a\right|=a\end{array}$

Calculation:

Two numbers is $2$ and $17$ .

Distance between the numbers can be calculated as shown below:

  $\begin{array}{c}d\left(a,b\right)=\left|a-b\right|\\ =\left|2-17\right|\\ =\left|-15\right|\end{array}$

Now, compare the distance with $\left|-a\right|$ :

  $\begin{array}{l}\left|-a\right|=\left|-15\right|\\ a=15\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|-15\right|=15$

Hence, distance between the numbers is 15 units.

b)

To determine

To find: the distance between the two numbers $-3$ and $21$ .

Answer to Problem 73E

The distance between the two numbers is $24$ .

Explanation of Solution

Given information:

Two numbers are given as

  $-3$ and $21$

Formula used:

Distance between two numbers a and b is given by

  $d\left(a,b\right)=\left|a-b\right|=\left|b-a\right|=d\left(b,a\right)$

Absolute value property:

For any real number a,

  $\begin{array}{l}\left|a\right|=a\ge 0\\ \left|-a\right|=\left|a\right|=a\end{array}$

Calculation:

Two numbers is $-3$ and $21$ .

Distance between the numbers can be calculated as shown below:

  $\begin{array}{c}d\left(a,b\right)=\left|a-b\right|\\ =\left|-3-21\right|\\ =\left|-24\right|\end{array}$

Now, compare the distance with $\left|-a\right|$ :

  $\begin{array}{l}\left|-a\right|=\left|-24\right|\\ a=24\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|-24\right|=24$

Hence, distance between the numbers is 24 units.

c)

To determine

To find: the distance between the two numbers $\frac{11}{8}$ and $-\frac{3}{10}$ .

Answer to Problem 73E

The distance between the two numbers is $\frac{67}{40}$ .

Explanation of Solution

Given information:

Two numbers are given as

  $\frac{11}{8}$ and $-\frac{3}{10}$

Formula used:

Distance between two numbers a and b is given by

  $d\left(a,b\right)=\left|a-b\right|=\left|b-a\right|=d\left(b,a\right)$

Absolute value property:

For any real number a,

  $\begin{array}{l}\left|a\right|=a\ge 0\\ \left|-a\right|=\left|a\right|=a\end{array}$

Calculation:

Two numbers is $\frac{11}{8}$ and $-\frac{3}{10}$ .

Distance between the numbers can be calculated as shown below:

  $\begin{array}{c}d\left(a,b\right)=\left|a-b\right|\\ =\left|\frac{11}{8}-\left(-\frac{3}{10}\right)\right|\\ =\left|\frac{11}{8}+\frac{3}{10}\right|\\ =\left|\frac{55+12}{40}\right|\\ =\left|\frac{67}{40}\right|\end{array}$

Now, compare the distance with $\left|a\right|$ :

  $\begin{array}{l}\left|a\right|=\left|\frac{67}{40}\right|\\ a=\frac{67}{40}\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|\frac{67}{40}\right|=\frac{67}{40}$

Hence, distance between the numbers is $\frac{67}{40}$ units.