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Chapter 9: Q3. (page 482)

Consider $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ if a, b, and c are real numbers. Determine whether each statement is sometimes, always, or never true. Explain your answer.
a. abc is a common denominator.
b. abc is the LCD.
c. ab is the LCD.
d. b is the LCD.
e. The sum is $\frac{b c + a c + a b}{a b c}$ .

Short Answer

Expert verified

Statements a, b and e are correct always true.

Statements c and d are sometimes true.

Step by step solution

01

Step 1. Given

Given rational expression is $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ .

02

Step 2. To determine

Determine whether each statement is sometimes, always, or never true. Explain your answer.

a. abc is a common denominator.

b. abc is the LCD.

c. ab is the LCD.

d. b is the LCD.

e. The sum is $\frac{b c + a c + a b}{a b c}$ .

03

Step 3. Calculation

The denominators of the rational expression $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ are a, b and c.

So, the LCM (least common multiple of them is abc.

We try to get abc as denominator for each term and then simplify:

$\begin{array}{l} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \\ = \frac{b c}{a b c} + \frac{a c}{a b c} + \frac{a b}{a b c} \\ = \frac{b c + a c + a b}{a b c} \end{array}$

It means statements a, b, and e are correct.

If b = c then the LCM of a, b, c = b will be ab. So, option c is sometimes true.

If a = b = c then LCM of a = b, b, c = b will be b. So, option d is sometimes true.

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