Simplify (2/(u+1)+4/(u-8))/(5/(u+1))

Solution:

Step:1

Invert the denominator and multiply it with the numerator: $\left(\frac{2}{u + 1} + \frac{4}{u - 8}\right) \cdot \frac{u + 1}{5}$

Step:2

Create a common denominator for $\frac{2}{u + 1}$ by multiplying it by $\frac{u - 8}{u - 8}$: $\left(\frac{2 \cdot \frac{u - 8}{u - 8}}{u + 1} + \frac{4}{u - 8}\right) \cdot \frac{u + 1}{5}$

Step:3

Create a common denominator for $\frac{4}{u - 8}$ by multiplying it by $\frac{u + 1}{u + 1}$: $\left(\frac{2 \cdot \frac{u - 8}{u - 8}}{u + 1} + \frac{4 \cdot \frac{u + 1}{u + 1}}{u - 8}\right) \cdot \frac{u + 1}{5}$

Step:4

Combine the fractions under a common denominator $(u + 1)(u - 8)$ by multiplying each by a suitable form of $1$.

Step:4.1

Multiply $\frac{2}{u + 1}$ by $\frac{u - 8}{u - 8}$: $\left(\frac{2(u - 8)}{(u + 1)(u - 8)} + \frac{4 \cdot \frac{u + 1}{u + 1}}{u - 8}\right) \cdot \frac{u + 1}{5}$

Step:4.2

Multiply $\frac{4}{u - 8}$ by $\frac{u + 1}{u + 1}$: $\left(\frac{2(u - 8)}{(u + 1)(u - 8)} + \frac{4(u + 1)}{(u + 1)(u - 8)}\right) \cdot \frac{u + 1}{5}$

Step:4.3

Arrange the factors of $(u - 8)(u + 1)$: $\left(\frac{2(u - 8)}{(u + 1)(u - 8)} + \frac{4(u + 1)}{(u + 1)(u - 8)}\right) \cdot \frac{u + 1}{5}$

Step:5

Combine the numerators over the common denominator: $\frac{2(u - 8) + 4(u + 1)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6

Simplify the numerator.

Step:6.1

Extract the common factor of $2$ from $2(u - 8) + 4(u + 1)$.

Step:6.1.1

Extract $2$ from $4(u + 1)$: $\frac{2(u - 8) + 2(2(u + 1))}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.1.2

Extract $2$ from $2(u - 8) + 2(2(u + 1))$: $\frac{2(u - 8 + 2(u + 1))}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.2

Apply the distributive property: $\frac{2(u - 8 + 2u + 2)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.3

Multiply $2$ by $1$: $\frac{2(u - 8 + 2u + 2)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.4

Combine $u$ and $2u$: $\frac{2(3u - 8 + 2)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.5

Combine $-8$ and $2$: $\frac{2(3u - 6)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.6

Extract the common factor of $3$ from $3u - 6$.

Step:6.6.1

Extract $3$ from $3u$: $\frac{2(3(u) - 6)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.6.2

Extract $3$ from $-6$: $\frac{2(3u + 3(-2))}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.6.3

Extract $3$ from $3u + 3(-2)$: $\frac{2(3(u - 2))}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:6.7

Multiply $2$ by $3$: $\frac{6(u - 2)}{(u + 1)(u - 8)} \cdot \frac{u + 1}{5}$

Step:7

Simplify the terms.

Step:7.1

Eliminate the common factor of $u + 1$.

Step:7.1.1

Cancel out the common factor: $\frac{6(u - 2)}{(u - 8)} \cdot \frac{1}{5}$

Step:7.1.2

Rewrite the simplified expression: $\frac{6(u - 2)}{u - 8} \cdot \frac{1}{5}$

Step:7.2

Multiply $\frac{6(u - 2)}{u - 8}$ by $\frac{1}{5}$: $\frac{6(u - 2)}{(u - 8) \cdot 5}$

Step:7.3

Reposition $5$ to the left of $u - 8$: $\frac{6(u - 2)}{5(u - 8)}$

Knowledge Notes:

The problem involves simplifying a complex fraction, which is a fraction where the numerator or the denominator (or both) are also fractions. The process includes finding a common denominator, combining fractions, and simplifying the result.

Key knowledge points include:

1. Multiplying by the reciprocal to divide fractions.

2. Finding a common denominator to combine fractions.

3. Simplifying expressions by factoring out common factors.

4. Applying the distributive property (a(b + c) = ab + ac).

5. Canceling common factors in the numerator and denominator.

6. Understanding that multiplying by a fraction equivalent to 1 does not change the value of an expression.

In this problem, we also see the importance of carefully manipulating algebraic expressions and the utility of factoring to simplify terms. It's crucial to ensure that all steps maintain the equivalence of the original expression to arrive at a simplified form.