Simplify ((x-4)/(x^2-x))/((x-1)/(2x^2+3x+1)-1/(x^2-1))

Solution:

Step:1

Take the reciprocal of the denominator and multiply it with the numerator: $\frac{x - 4}{x^{2} - x} \cdot \frac{1}{\frac{x - 1}{2x^{2} + 3x + 1} - \frac{1}{x^{2} - 1}}$

Step:2

Extract $x$ from $x^{2} - x$.

Step:2.1

Extract $x$ from $x^{2}$: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{2x^{2} + 3x + 1} - \frac{1}{x^{2} - 1}}$

Step:3

Use the grouping method to factor.

Step:3.1

For $ax^{2} + bx + c$, split the middle term into two terms whose product is $ac = 2 \cdot 1 = 2$ and sum is $b = 3$.

Step:3.1.1

Extract $3$ from $3x$: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{2x^{2} + 3(x) + 1} - \frac{1}{x^{2} - 1}}$

Step:3.1.2

Rewrite $3$ as $1 + 2$: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{2x^{2} + (1 + 2)x + 1} - \frac{1}{x^{2} - 1}}$

Step:3.1.3

Apply the distributive property: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{2x^{2} + x + 2x + 1} - \frac{1}{x^{2} - 1}}$

Step:3.2

Factor out the greatest common factor from each group.

Step:3.2.1

Group the first two and the last two terms: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{(2x^{2} + x) + (2x + 1)} - \frac{1}{x^{2} - 1}}$

Step:3.2.2

Factor out the GCF from each group: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{x(2x + 1) + (2x + 1)} - \frac{1}{x^{2} - 1}}$

Step:4

Simplify the denominator.

Step:4.1

Rewrite $1$ as $1^{2}$: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{(2x + 1)(x + 1)} - \frac{1}{x^{2} - 1^{2}}}$

Step:4.2

Factor using the difference of squares formula: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{x - 1}{(2x + 1)(x + 1)} - \frac{1}{(x + 1)(x - 1)}}$

Step:5

Simplify the denominator.

Step:5.1

Multiply by $\frac{x - 1}{x - 1}$ to get a common denominator: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{(x - 1)^{2}}{(2x + 1)(x + 1)(x - 1)} - \frac{1}{(x + 1)(x - 1)}}$

Step:5.2

Multiply by $\frac{2x + 1}{2x + 1}$ to get a common denominator: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{1}{\frac{(x - 1)^{2}}{(2x + 1)(x + 1)(x - 1)} - \frac{2x + 1}{(2x + 1)(x + 1)(x - 1)}}$

Step:6

Multiply the numerator by the reciprocal of the denominator: $\frac{x - 4}{x \cdot (x - 1)} \cdot \frac{(2x + 1)(x + 1)(x - 1)}{(x - 1)^{2} - (2x + 1)}$

Step:7

Cancel the common factors.

Step:7.1

Cancel $x - 4$: $\frac{1}{x \cdot (x - 1)} \cdot \frac{(2x + 1)(x + 1)(x - 1)}{x}$

Step:7.2

Cancel $x - 1$: $\frac{1}{x} \cdot \frac{(2x + 1)(x + 1)}{x}$

Step:8

Multiply $\frac{1}{x}$ by $\frac{(2x + 1)(x + 1)}{x}$: $\frac{(2x + 1)(x + 1)}{x^{2}}$

Knowledge Notes:

1. Reciprocal Multiplication: When dividing fractions, multiply by the reciprocal of the divisor.

2. Factoring: Extracting common factors from terms to simplify expressions.

3. Grouping Method: A factoring technique that involves rearranging terms and factoring in groups.

4. Difference of Squares: A formula used to factor expressions in the form of $a^{2} - b^{2} = (a + b)(a - b)$.

5. Common Denominator: When combining fractions, a common denominator is required. This often involves multiplying by a form of one to achieve the same denominator.

6. Canceling Common Factors: When the same factor appears in both the numerator and denominator, it can be canceled out.

7. Power Rule: For exponents, $a^{m} \cdot a^{n} = a^{m+n}$.

8. Distributive Property: Multiplying a single term across terms within parentheses, such as $a(b + c) = ab + ac$.