Question

Simplifying Rational Expressions: I need answers for both 7 and 8 below. Answers for just one or the other is also fine.

Answer 1

Answer:

1. Option A

2. Option D

Step by step explanation

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

Use $\frac{ - a}{b} = \frac{a}{ - b} = - \frac{a}{b}$ to rewrite the fractions

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

Write all numerators above the Least Common Denominator ( X - 1 ) ( X + 1 )

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

When there is a ( - ) in front of an expression in parentheses , change the sign of each term in the expression

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

Using $(a - b)(a + b) = {a}^{2} - {b}^{2}$ , simplify the product

$\frac{ - x - 1 + x}{ {x}^{2} - 1 }$

Since two opposites add up to zero, remove them from the expression

$\frac{ - 1}{ {x}^{2} - 1}$

So, Option A is the right option.

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2.

$\frac{ {x}^{2} - x - 12}{ {x}^{2} - 16} - \frac{1 - 2x}{x + 4}$

Write - X as a difference

$\frac{ {x}^{2} + 3x - 4x - 12 }{ {x}^{2} - 16 } - \frac{1 - 2x}{x + 4}$

Using ${a}^{2} - {b}^{2} = (a - b)(a + b)$ , factor the expression

$\frac{ {x}^{2} + 3x - 4x - 12 }{(x - 4)(x + 4)} - \frac{1 - 2x}{x + 4}$

Factor the expression

$\frac{x(x + 3) - 4(x + 3)}{(x - 4)(x + 4)} - \frac{1 - 2x}{x + 4}$

Factor out X+3 from the expression

$\frac{(x + 3)(x - 4)}{(x - 4)(x + 4)} - \frac{1 - 2x}{x + 4}$

Reduce the fraction with x-4

$\frac{x + 3}{x + 4} - \frac{1 - 2x}{x + 4}$

Write all the numerators above the common denominator

$\frac{x + 3 - ( 1- 2x)}{x + 4}$

When there is a (-) in front of an expression in parentheses, change the sign of each term in the expression

$\frac{x + 3 - 1 + 2x}{x + 4}$

Collect like terms

$\frac{3x + 3 - 1}{x + 4}$

Subtract the numbers

$\frac{3x + 2}{x + 4}$

Undefined at,

X + 4 = 0

Move constant to R.H.S and change its sign

$x = 0 - 4$

Calculate

$x = - 4$

So, the answer is :

$\frac{3x + 2}{x + 4}$ , undefined at X = -4 and 4

Hope this helps..

Best regards!!