(2(x−2)) + 1/(6(x−2)) − 1/(3(x−1)) (x+1) + 2/((x−1)2) − x2/((x−1)3)

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the given expressions.

:

\frac{1}{2(x-2)} + \frac{1}{6(x-2)} - \frac{1}{3(x-1)}

Find the least common denominator (LCD).
The denominators are $2(x-2)$ , $6(x-2)$ , and $3(x-1)$ . The LCD is $6(x-2)(x-1)$ .
Rewrite each fraction with the LCD.
$\frac{1 \cdot 3(x-1)}{2(x-2) \cdot 3(x-1)} + \frac{1 \cdot (x-1)}{6(x-2) \cdot (x-1)} - \frac{1 \cdot 2(x-2)}{3(x-1) \cdot 2(x-2)}$ $= \frac{3(x-1)}{6(x-2)(x-1)} + \frac{x-1}{6(x-2)(x-1)} - \frac{2(x-2)}{6(x-2)(x-1)}$
Combine the numerators over the common denominator.
$= \frac{3(x-1) + (x-1) - 2(x-2)}{6(x-2)(x-1)}$
Simplify the numerator.
$= \frac{3x - 3 + x - 1 - 2x + 4}{6(x-2)(x-1)}$ $= \frac{(3x + x - 2x) + (-3 - 1 + 4)}{6(x-2)(x-1)}$ $= \frac{2x + 0}{6(x-2)(x-1)}$ $= \frac{2x}{6(x-2)(x-1)}$
Reduce the fraction.
$= \frac{x}{3(x-2)(x-1)}$
The simplified expression is:
$\boxed{\frac{x}{3(x-2)(x-1)}}$
:
$\frac{1}{x+1} + \frac{2}{(x-1)^2} - \frac{x^2}{(x-1)^3}$
Find the least common denominator (LCD).
The denominators are $(x+1)$ , $(x-1)^2$ , and $(x-1)^3$ . The LCD is $(x+1)(x-1)^3$ .
Rewrite each fraction with the LCD.
$\frac{1 \cdot (x-1)^3}{(x+1)(x-1)^3} + \frac{2 \cdot (x+1)(x-1)}{(x-1)^2 \cdot (x+1)(x-1)} - \frac{x^2 \cdot (x+1)}{(x-1)^3 \cdot (x+1)}$ $= \frac{(x-1)^3}{(x+1)(x-1)^3} + \frac{2(x^2-1)}{(x+1)(x-1)^3} - \frac{x^2(x+1)}{(x+1)(x-1)^3}$
Combine the numerators over the common denominator.
$= \frac{(x-1)^3 + 2(x^2-1) - x^2(x+1)}{(x+1)(x-1)^3}$
Expand and simplify the numerator.
Recall $(x-1)^3 = x^3 - 3x^2 + 3x - 1$ .
$= \frac{(x^3 - 3x^2 + 3x - 1) + (2x^2 - 2) - (x^3 + x^2)}{(x+1)(x-1)^3}$ $= \frac{x^3 - 3x^2 + 3x - 1 + 2x^2 - 2 - x^3 - x^2}{(x+1)(x-1)^3}$
Combine like terms:
$= \frac{(x^3 - x^3) + (-3x^2 + 2x^2 - x^2) + (3x) + (-1 - 2)}{(x+1)(x-1)^3}$ $= \frac{0 - 2x^2 + 3x - 3}{(x+1)(x-1)^3}$ $= \frac{-2x^2 + 3x - 3}{(x+1)(x-1)^3}$
The simplified expression is:
$\boxed{\frac{-2x^2 + 3x - 3}{(x+1)(x-1)^3}}$

(2(x-2)) + 1/(6(x-2)) - 1/(3(x-1)) (x+1) + 2/((x-1)2) - x2/((x-1)3)

Answer

Step-by-step solution

Find the least common denominator (LCD).

Rewrite each fraction with the LCD.

Combine the numerators over the common denominator.

Simplify the numerator.

Reduce the fraction.

Find the least common denominator (LCD).

Rewrite each fraction with the LCD.

Combine the numerators over the common denominator.

Expand and simplify the numerator.

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