msurls wrote:

Bunuel wrote:

Which of the following is equivalent to \(\frac{1 - \frac{1}{x + 1}}{x}\) for all values of x for which both expressions are defined?

(A) 1

(B) x + 1

(C) 1/x

(D) 1/(x + 1)

(E) x^2 + x

I got this one by choosing smart number. But I don't know how to solve it algebraically. Can someone please write out the steps? Thanks.

msurls , sure. (Just making sure that you saw

niks18 's solution.) Complex fractions can be a pain. \(\frac{1 - \frac{1}{x + 1}}{x}\)

One method (and there are many): simplify the numerator first, then the whole fraction

If there were a fraction in the denominator (instead of in the numerator, as here),

you could simplify the denominator first.

Fractions in both numerator and denominator?

Typically, use LCM of all fractions in the complex fraction, and

multiply that LCM "through" all terms in numerator and denominator.

Here:

1) Simplify numerator ONLY. Use LCM to convert terms into like fractions. Numerator:

\(1 -\frac{1}{x+1}\)LCM of 1 and (x+1) is (x+1)

Second term has correct denominator. Multiply first term by

\(\frac{LCM}{LCM}\)(equivalent of 1)

\((\frac{x+1}{x+1} * 1) - \frac{1}{x+1}\) =

\(\frac{x+1}{x+1} - \frac{1}{x+1}\)

2) Subtract (denominator does not change)

\((\frac{x+1}{x+1} - \frac{1}{x+1}) =\frac{(x+1)-1}{x+1}=\frac{x}{x+1}\)

3) Put that whole expression, the new numerator, over the original denominator, \(x\)

\(\frac{(\frac{x}{x+1})}{x}\)

4) Express division as multiplication* and multiply

\(\frac{(\frac{x}{x+1})}{x} =(\frac{x}{x+1}*\frac{1}{x})= \frac{x*1}{(x+1)*x}=\frac{x}{x(x+1)}\)

5) Simplify. Factor out ("cancel") \(x\) in numerator and denominator:

\(\frac{x}{x(x+1)}=\frac{1}{(x+1)}\)

Answer D

* Division as multiplication

\(\frac{(\frac{a}{b})}{c}=\frac{a}{b}*\frac{1}{c}=\frac{a}{bc}\)

In short, \(\frac{(\frac{a}{b})}{c}=\frac{a}{b*c}\)Hope that helps.

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