Question

Add, subtract, multiply, or divide as indicated. List any restrictions for the variable(s) and simplify the answers when possible.

1) x-2 / x+5 + 3x / 2x-1

2) x+6 / x-6 - x^2 / x+6

3) x+9 / x-4 + x+2 / x^2-11x+28

4) x / x^2-64 + 11 / 2x^2+11x-40

5) 5 / x + 11 / x-3 - x-4 / x^2+2x-15

Answer 1

QUESTION 6

We want to simplify;

$\frac{x - 2}{x + 5} + \frac{3x}{2x - 1}$

We collect LCM to get,



$= \frac{(x - 2)(2x - 1) + 3x(x + 5)}{(x + 5)(2x - 1)}$




Expand the numerator to get;

$= \frac{2 {x}^{2} - x - 4x + 2+ 3 {x}^{2} + 15x}{(x + 5)(2x - 1)}$


$= \frac{2 {x}^{2} +3 {x}^{2}- x - 4x +15x + 2 }{(x + 5)(2x - 1)}$


$= \frac{5{x}^{2} + 10x + 2 }{(x + 5)(2x - 1)}$


QUESTION 7


The given expression is
$\frac{x + 6}{x - 6} - \frac{ {x}^{2} }{x + 6}$


$= \frac{(x + 6)(x - 6) - {x}^{2}(x - 6) }{(x - 6)(x + 6)}$


Expand the bracket to obtain,

$= \frac{{x}^{2} - 36 - {x}^{3} + 6 {x}^{2} }{(x - 6)(x + 6)}$

This simplifies to

$= \frac{ - {x}^{3} + 7{x}^{2} - 36 }{(x - 6)(x + 6)}$


QUESTION 8


We want to simplify

$\frac{x + 9}{x - 4} + \frac{x + 2}{ {x}^{2} - 11x + 28 }$

Let us factor the denominator of the fraction first.



$= \frac{x + 9}{x - 4} + \frac{x + 2}{ {x}^{2} - 7x - 4x+ 28 }$




$= \frac{x + 9}{x - 4} + \frac{x + 2}{ x(x - 7) - 4(x - 7)}$



$= \frac{x + 9}{x - 4} + \frac{x + 2}{ (x - 7)(x- 4)}$



We collect LCM to obtain;


$= \frac{(x + 9)(x - 7) + (x + 2)}{ (x - 7)(x- 4)} .$


We expand brackets to get;

$= \frac{ {x}^{2} - 7x + 9x - 63+ x + 2}{ (x - 7)(x- 4)} .$

$= \frac{ {x}^{2} + 3x - 61}{ (x - 7)(x- 4)} .$



QUESTION 9

The given expression is

$\frac{x}{ {x}^{2} - 64} + \frac{11}{2 {x}^{2} + 11x - 40}$




We factor the numerator of the second fraction to get,


$\frac{x}{ {x}^{2} - {8}^{2} } + \frac{11}{2 {x}^{2} + 16x - 5x- 40}$



$= \frac{x}{ {x}^{2} - {8}^{2} } + \frac{11}{2 x(x + 8) - 5(x + 8)}$


This implies that,

$= \frac{x}{ (x - 8)( x + 8)} + \frac{11}{(2 x - 5)(x + 8)} .$


We collect LCM to get,

$= \frac{x(2x - 5) + 11(x - 8)}{(2 x - 5)(x + 8)(x - 8)}$


$= \frac{2 {x}^{2} - 5x + 11x - 88}{(2 x - 5)(x + 8)(x - 8)}$


$= \frac{2 {x}^{2} + 6x - 88}{(2 x - 5)(x + 8)(x - 8)}$

QUESTION 10

The given expression is

$\frac{5}{x} + \frac{11}{x - 3} - \frac{x - 4}{ {x}^{2} + 2x - 15}$


We factor the denominator to obtain:



$\frac{5}{x} + \frac{11}{x - 3} - \frac{x - 4}{ (x - 3)(x + 5)}$


We collect LCM to get;

$\frac{5(x - 3)(x + 5) + 11x(x + 5) - x(x - 4)}{ x(x - 3)(x + 5)}$

We expand brackets to get,


$\frac{5 {x}^{2} + 10x - 75 + 11 {x}^{2} + 55x- {x}^{2} + 4x}{ x(x - 3)(x + 5)}$


$\frac{15 {x}^{2} + 69x - 75 }{ x(x - 3)(x + 5)}$