Chapter 1.1, Problem 34E

(a)

To determine

To state: whether the inequality $\frac{10}{11}<\frac{12}{13}$ is true or false.

Answer to Problem 34E

  $\frac{10}{11}<\frac{12}{13}$ is true.

Explanation of Solution

Given information:

An inequality as $\frac{10}{11}<\frac{12}{13}$ .

Formula used:

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ .

  $\frac{a}{b}<\frac{c}{d}$ if $a<c,b=d$

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ .

  $\frac{a}{b}>\frac{c}{d}$ if $a>c,b=d$

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ .

  $\frac{a}{b}=\frac{c}{d}$ if $a=c,b=d$

Calculation:

Consider the given inequality.

  $\frac{10}{11}<\frac{12}{13}$

Now, multiply numerator and denominator of $\frac{10}{11}$ by 13 and $\frac{12}{13}$ by 11.

  $\begin{array}{l}\frac{10}{11}=\frac{10\times 13}{13\times 11}=\frac{130}{143}\\ \frac{12}{13}=\frac{12\times 11}{13\times 11}=\frac{132}{143}\end{array}$

Here, for $\frac{130}{143}$ and $\frac{132}{143}$ ,

  $130<132$

Hence, $\frac{130}{143}<\frac{132}{143}$

Hence, the inequality $\frac{10}{11}<\frac{12}{13}$ is true.

(b)

To determine

To state: whether the inequality $-\frac{1}{2}<-1$ is true or false.

Answer to Problem 34E

  $-\frac{1}{2}<-1$ is true.

Explanation of Solution

Given information:

An inequality as $-\frac{1}{2}<-1$ .

Formula used:

For two numbers a and b

  $-a<-b$ if $a>b$

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ .

  $\frac{a}{b}<\frac{c}{d}$ if $a<c,b=d$

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ .

  $\frac{a}{b}>\frac{c}{d}$ if $a>c,b=d$

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$ .

  $\frac{a}{b}=\frac{c}{d}$ if $a=c,b=d$

Calculation:

Consider the given inequality.

  $-\frac{1}{2}<-1$

Now, multiply numerator and denominator of $1$ by 2.

  $1=\frac{1\times 2}{1\times 2}=\frac{2}{2}$

Here, for $\frac{1}{2}$ and $\frac{2}{2}$ ,

  $1<2$

Hence, $\frac{1}{2}<\frac{2}{2}$

Multiply both sides by $\left(-1\right)$ .

  $\begin{array}{c}\frac{1}{2}\left(-1\right)<\frac{2}{2}\left(-1\right)\\ -\frac{1}{2}>-\frac{2}{2}\\ -\frac{1}{2}>-1\end{array}$

Hence, the inequality $-\frac{1}{2}<-1$ is false.