Simplifying rational expressions

[humm35555]

I have a few algebra problems I need help with & I need to see how to work it out:
I am so confused.

1) Simplify (2x + 6)/(x-1) divided by (x^2 + x - 6)/(x^2 - 1)

2) Simplify (x + 1)/(x - 5) - (x - 3)/(x + 5)
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Edited by stapel -- Reason for edit: Please don't edit post after you've received a reply, to add new questions. New questions belong in new threads. Please show some appreciation for the tutors. Thank you.

 

[soroban]

Re: algebra help please...

Hello, humm35555!

I'll walk through these . . .

\(\displaystyle \L1)\:\frac{2x\,+\,6}{x\,-\,1}\,\div\,\frac{x^2\,+\,x\,-\,6}{x^2\,-\,1}\)

To divide by a fraction, invert the divisor (second fraction) and multiply.

So we have: \(\displaystyle \L\:\frac{2x\,+\,6}{x\,-\,1}\,\cdot\,\frac{x^2\,-\,1}{x^2\,+\,x\,-\,6}\)


To multiply fractions, factor as much as possible:

. . . . \(\displaystyle \L\frac{2(x\,+\,3)}{x\,-\,1}\,\cdot\,\frac{(x\,-\,1)(x\,+\,1)}{(x\,-\,2)(x\,+\,3)\)


Then cancel common factors in the numerator and the denominator:

. . . . \(\displaystyle \L\frac{2\sout{(x\,+\,3)}}{\sout{x\,-\,1}}\,\cdot\,\frac{\sout{(x\,-\,1)}(x\,+\,1)}{(x\,-\,2)\sout{(x\,+\,3)}}\)


And "copy" what is left: \(\displaystyle \L\:\frac{2(x\,+\,1)}{x\,-\,2}\)

\(\displaystyle \L2)\;\frac{x\,+\,1}{x\,-\,5}\,-\,\frac{x\,-\,3}{x\,+\,5}\)

To add or subtract fractions, we must get a common denominator.
. . The LCD is: \(\displaystyle (x\,-\,5)(x\,+\,5)\)

Multiply top and bottom of each fraction by an appropriate expression
. . to convert each fraction to the LCD.

\(\displaystyle \L\frac{x\,+\,1}{x\,-\,5}\,\cdot\,\frac{x\,+\,5}{x\,+\,5}\:-\:\frac{x\,-\,3}{x\,+\,5}\,\cdot\frac{x\,-\,5}{x\,-\,5} \;=\; \frac{(x\,+\,1)(x\,+\,5)}{(x\,-\,5)(x\,+\,5)}\:-\:\frac{(x\,-\,3)(x\,-\,5)}{(x\,-\,5)(x\,+\,5)}\)


The two fractions has a common denominator.
. . We can combine the numerators:

. . \(\displaystyle \L\frac{(x\,+\,1)(x\,+\,5) \:-\x\,-\,3)(x\,-\,5)}{(x\,-\,5)(x\,+\,5)} \;=\; \frac{(x^2\,+\,6x\,+\,5)\,-\,(x^2\,-\,8x\,+\,15)}{(x\,-\,5)(x\,+\,5)}\)

. . \(\displaystyle \L=\;\frac{x^2\,+\,6x\,+\,5\,-\,x^2\,+\,8x\,-\,15}{(x\,-\,5)(x\,+\,5)} \;=\;\frac{14x\,-\,10}{(x\,-\,5)(x\,+\,5)}\)