## Simplifying expression: x - (x - 1) / [x / (x + 1)]

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.
Beini
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### Simplifying expression: x - (x - 1) / [x / (x + 1)]

$x - \frac{x-1}{\frac{x}{x+1}}$

The book gives an answer of $\frac{1}{x}$. This is probably very simple problem but I really don't understand how to workout a problem like that.

I would appreciate if anyone would like to show the steps involved getting to that answer and maybe point to a related lesson

stapel_eliz
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Beini wrote:I really don't understand how to workout a problem like that.

To learn, please try this lesson on "complex" fractions and this one on adding (and subtracting) rational expressions.

First, simplify the complex fraction (the part after the initial "x minus"). A good first step for this will be to multiply, top and bottom, by "x + 1":

. . . . .$\frac{\left(\frac{x\, -\, 1}{1}\right)}{\left(\frac{x}{x\, +\, 1}\right)}\frac{\left(\frac{x\, +\, 1}{1}\right)}{\left(\frac{x\, +\, 1}{1}\right)}\, =\, \frac{x^2\, -\, 1}{x}$

Then convert the linear term into fractional form, having a common denominator. Then combine and simplify.

MrAlgebra
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### Re: Simplifying expression: x - (x - 1) / [x / (x + 1)]

Beini wrote:$x - \frac{x-1}{\frac{x}{x+1}}$

The book gives an answer of $\frac{1}{x}$. This is probably very simple problem but I really don't understand how to workout a problem like that.

I would appreciate if anyone would like to show the steps involved getting to that answer and maybe point to a related lesson

It's been a few days so I'm going to work the problem out how I would do it. I would start by getting rid of the complex fraction by flipping the bottom and multiplying.

$x - \frac{x-1}{\frac{x}{x+1}}$ The starting equation.

$x - (x-1)(\frac{x+1}{x})$ Flip the bottom and multiply.

$x - \frac{(x-1)(x+1)}{x}$ Multiply the second term.

Then I would get a common denominator, which will be x:

$x(\frac{x}{x}) - \frac{(x-1)(x+1)}{x}$ Multiplying the first term by (x/x) to get the common denominator.

$\frac{x^2}{x} - \frac{(x-1)(x+1)}{x}$ Done the multiplication.

$\frac{x^2 - (x-1)(x+1)}{x}$ Combined the fractions for subtraction since we have a common denominator.

$\frac{x^2 - (x^2-1)}{x}$ Difference of squares identity where (x+1)(x-1) = x² - 1.

$\frac{x^2 - x^2 + 1}{x}$ Distribute the negative.

$\frac{1}{x}$ Simplify the numerator.

Ta da!