Chapter 1.1, Problem 74E

a)

To determine

To find: the distance between the two numbers $\frac{7}{15}$ and $-\frac{1}{21}$ .

Answer to Problem 74E

The distance between the two numbers is $15$ .

Explanation of Solution

Given information:

Two numbers are given as

  $\frac{7}{15}$ and $-\frac{1}{21}$ .

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 $\frac{7}{15}$ and $-\frac{1}{21}$ .

Distance between the numbers can be calculated as shown below:

  $\begin{array}{c}d\left(a,b\right)=\left|a-b\right|\\ =\left|\frac{7}{15}-\left(-\frac{1}{21}\right)\right|\\ =\left|\frac{7}{15}+\frac{1}{21}\right|\\ =\left|\frac{49+5}{105}\right|\\ =\left|\frac{54}{105}\right|\end{array}$

Now, compare the distancewith $\left|a\right|$ :

  $\begin{array}{l}\left|-a\right|=\left|\frac{54}{105}\right|\\ a=\frac{54}{105}\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|\frac{54}{105}\right|=\frac{54}{105}$

Hence, distance between the numbers is $\frac{54}{105}$ units.

b)

To determine

To find: the distance between the two numbers $-38$ and $-57$ .

Answer to Problem 74E

The distance between the two numbers is $21$ .

Explanation of Solution

Given information:

Two numbers are given as

  $-38$ and $-57$

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 $-38$ and $-57$

Distance between the numbers can be calculated as shown below:

  $\begin{array}{c}d\left(a,b\right)=\left|a-b\right|\\ =\left|-38-\left(-57\right)\right|\\ =\left|-38+57\right|\\ =\left|21\right|\end{array}$

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

  $\begin{array}{l}\left|-a\right|=\left|21\right|\\ a=21\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|21\right|=21$

Hence, distance between the numbers is 21 units.

c)

To determine

To find: the distance between the two numbers $-2.6$ and $-1.8$ .

Answer to Problem 74E

The distance between the two numbers is $44$ .

Explanation of Solution

Given information:

Two numbers are given as

  $-2.6$ and $-1.8$

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 $-2.6$ and $-1.8$ .

Distance between the numbers can be calculated as shown below:

  $\begin{array}{c}d\left(a,b\right)=\left|a-b\right|\\ =\left|-2.6-\left(-1.8\right)\right|\\ =\left|2.6+1.8\right|\\ =\left|44\right|\end{array}$

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

  $\begin{array}{l}\left|a\right|=\left|44\right|\\ a=44\end{array}$

Now, by absolute value property:

  $\left|a\right|=a\ge 0$

Therefore, $\left|44\right|=44$

Hence, distance between the numbers is $44$ units.