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.