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

1. Option A

2. Option D

Step by step explanation

1.
`(1)/(1 - x) + \frac{x}{ {x}^(2) - 1}`

Use
`( - a)/(b) = (a)/( - b) = - (a)/(b)` to rewrite the fractions

`- (1)/(x - 1) + (x)/((x - 1)(x + 1))`

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

`( - (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

`( - 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.

___________________________________

2.

`\frac{ {x}^(2) - x - 12}{ {x}^(2) - 16} - (1 - 2x)/(x + 4)`

Write - X as a difference

`\frac{ {x}^(2) + 3x - 4x - 12 }{ {x}^(2) - 16 } - (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)} - (1 - 2x)/(x + 4)`

Factor the expression

`(x(x + 3) - 4(x + 3))/((x - 4)(x + 4)) - (1 - 2x)/(x + 4)`

Factor out X+3 from the expression

`((x + 3)(x - 4))/((x - 4)(x + 4)) - (1 - 2x)/(x + 4)`

Reduce the fraction with x-4

`(x + 3)/(x + 4) - (1 - 2x)/(x + 4)`

Write all the numerators above the common denominator

`(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

`(x + 3 - 1 + 2x)/(x + 4)`

Collect like terms

`(3x + 3 - 1)/(x + 4)`

Subtract the numbers

`(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 :

`(3x + 2)/(x + 4)` , undefined at X = -4 and 4

Hope this helps..

Best regards!!