Simplify ((4x-16)/(5x+15))/((4-x)/(2x+6))

Solution:

Step:1

Take the inverse of the denominator and multiply it with the numerator: $\frac{4x - 16}{5x + 15} \cdot \frac{2x + 6}{4 - x}$

Step:2

Begin simplification process.

Step:2.1

Extract the common factor from the first term in the numerator.

Step:2.1.1

Pull out the common factor from $4x$: $\frac{4(x) - 16}{5x + 15} \cdot \frac{2x + 6}{4 - x}$

Step:2.1.2

Extract the common factor from $-16$: $\frac{4x - 4 \cdot 4}{5x + 15} \cdot \frac{2x + 6}{4 - x}$

Step:2.1.3

Combine the factored terms: $\frac{4(x - 4)}{5x + 15} \cdot \frac{2x + 6}{4 - x}$

Step:2.2

Identify and factor out the common factor from the first term in the denominator.

Step:2.2.1

Isolate the common factor from $5x$: $\frac{4(x - 4)}{5(x) + 15} \cdot \frac{2x + 6}{4 - x}$

Step:2.2.2

Pull out the common factor from $15$: $\frac{4(x - 4)}{5x + 5 \cdot 3} \cdot \frac{2x + 6}{4 - x}$

Step:2.2.3

Consolidate the factored terms: $\frac{4(x - 4)}{5(x + 3)} \cdot \frac{2x + 6}{4 - x}$

Step:2.3

Extract the common factor from the second term in the numerator.

Step:2.3.1

Isolate the common factor from $2x$: $\frac{4(x - 4)}{5(x + 3)} \cdot \frac{2(x) + 6}{4 - x}$

Step:2.3.2

Extract the common factor from $6$: $\frac{4(x - 4)}{5(x + 3)} \cdot \frac{2x + 2 \cdot 3}{4 - x}$

Step:2.3.3

Combine the factored terms: $\frac{4(x - 4)}{5(x + 3)} \cdot \frac{2(x + 3)}{4 - x}$

Step:2.4

Eliminate the common factor between the numerator and the denominator.

Step:2.4.1

Factor out $(x + 3)$ from the denominator: $\frac{4(x - 4)}{(x + 3) \cdot 5} \cdot \frac{2(x + 3)}{4 - x}$

Step:2.4.2

Factor out $(x + 3)$ from the numerator: $\frac{4(x - 4)}{(x + 3) \cdot 5} \cdot \frac{(x + 3) \cdot 2}{4 - x}$

Step:2.4.3

Cancel out the common $(x + 3)$ factor: $\frac{4(x - 4)}{(\cancel{x + 3}) \cdot 5} \cdot \frac{(\cancel{x + 3}) \cdot 2}{4 - x}$

Step:2.4.4

Simplify the expression: $\frac{4(x - 4)}{5} \cdot \frac{2}{4 - x}$

Step:2.5

Perform the multiplication of the simplified fractions: $\frac{4(x - 4) \cdot 2}{5(4 - x)}$

Step:2.6

Multiply the constants: $\frac{8(x - 4)}{5(4 - x)}$

Step:2.7

Simplify by canceling out the common factor $(x - 4)$ and $(4 - x)$.

Step:2.7.1

Factor out $-1$ from $x$: $\frac{8(-1(-x) - 4)}{5(4 - x)}$

Step:2.7.2

Express $-4$ as $-1(4)$: $\frac{8(-1(-x) - 1(4))}{5(4 - x)}$

Step:2.7.3

Factor out $-1$ from the expression: $\frac{8(-1(-x + 4))}{5(4 - x)}$

Step:2.7.4

Rearrange the terms: $\frac{8(-1(-x + 4))}{5(-x + 4)}$

Step:2.7.5

Cancel out the common factor: $\frac{8(-1(\cancel{-x + 4}))}{5(\cancel{-x + 4})}$

Step:2.7.6

Finalize the expression: $\frac{8 \cdot (-1)}{5}$

Step:3

Multiply the constants: $\frac{-8}{5}$

Step:4

Place the negative sign in front of the fraction: $-\frac{8}{5}$

Step:5

Present the result in various forms:

Exact Form: $-\frac{8}{5}$ Decimal Form: $-1.6$ Mixed Number Form: $-1\frac{3}{5}$

Knowledge Notes:

The problem involves simplifying a complex fraction by performing operations such as factoring, multiplying by the reciprocal, and canceling common factors. Here are the relevant knowledge points:

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

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

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

4. Negative Numbers: Factoring out $-1$ can help in simplifying expressions with negative terms.

5. Simplifying Complex Fractions: Complex fractions can be simplified by performing multiplication and cancellation step by step.

6. Representation of Rational Numbers: Rational numbers can be represented in exact form, decimal form, and mixed number form.