Solve (4/x-3+3)(10/x(x-2)+4/x-5/x-2)

$$( \frac { 4 } { x - 3 } + 3 ) ( \frac { 10 } { x ( x - 2 ) } + \frac { 4 } { x } - \frac { 5 } { x - 2 } )$$

Evaluate

$$-\frac{3x-5}{x\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-3}{x-3}.

$$\left(\frac{4}{x-3}+\frac{3\left(x-3\right)}{x-3}\right)\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

Since \frac{4}{x-3} and \frac{3\left(x-3\right)}{x-3} have the same denominator, add them by adding their numerators.

$$\frac{4+3\left(x-3\right)}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

Do the multiplications in 4+3\left(x-3\right).

$$\frac{4+3x-9}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

Combine like terms in 4+3x-9.

$$\frac{-5+3x}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right) and x is x\left(x-2\right). Multiply \frac{4}{x} times \frac{x-2}{x-2}.

$$\frac{-5+3x}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4\left(x-2\right)}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

Since \frac{10}{x\left(x-2\right)} and \frac{4\left(x-2\right)}{x\left(x-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-5+3x}{x-3}\left(\frac{10+4\left(x-2\right)}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

Do the multiplications in 10+4\left(x-2\right).

$$\frac{-5+3x}{x-3}\left(\frac{10+4x-8}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

Combine like terms in 10+4x-8.

$$\frac{-5+3x}{x-3}\left(\frac{2+4x}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right) and x-2 is x\left(x-2\right). Multiply \frac{5}{x-2} times \frac{x}{x}.

$$\frac{-5+3x}{x-3}\left(\frac{2+4x}{x\left(x-2\right)}-\frac{5x}{x\left(x-2\right)}\right)$$

Since \frac{2+4x}{x\left(x-2\right)} and \frac{5x}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-5+3x}{x-3}\times \frac{2+4x-5x}{x\left(x-2\right)}$$

Combine like terms in 2+4x-5x.

$$\frac{-5+3x}{x-3}\times \frac{2-x}{x\left(x-2\right)}$$

Extract the negative sign in 2-x.

$$\frac{-5+3x}{x-3}\times \frac{-\left(x-2\right)}{x\left(x-2\right)}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{-5+3x}{x-3}\times \frac{-1}{x}$$

Multiply \frac{-5+3x}{x-3} times \frac{-1}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-5+3x\right)\left(-1\right)}{\left(x-3\right)x}$$

Use the distributive property to multiply -5+3x by -1.

$$\frac{5-3x}{\left(x-3\right)x}$$

Use the distributive property to multiply x-3 by x.

$$\frac{5-3x}{x^{2}-3x}$$

Expand

$$-\frac{3x-5}{x\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-3}{x-3}.

$$\left(\frac{4}{x-3}+\frac{3\left(x-3\right)}{x-3}\right)\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

Since \frac{4}{x-3} and \frac{3\left(x-3\right)}{x-3} have the same denominator, add them by adding their numerators.

$$\frac{4+3\left(x-3\right)}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

Do the multiplications in 4+3\left(x-3\right).

$$\frac{4+3x-9}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

Combine like terms in 4+3x-9.

$$\frac{-5+3x}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4}{x}-\frac{5}{x-2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right) and x is x\left(x-2\right). Multiply \frac{4}{x} times \frac{x-2}{x-2}.

$$\frac{-5+3x}{x-3}\left(\frac{10}{x\left(x-2\right)}+\frac{4\left(x-2\right)}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

Since \frac{10}{x\left(x-2\right)} and \frac{4\left(x-2\right)}{x\left(x-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-5+3x}{x-3}\left(\frac{10+4\left(x-2\right)}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

Do the multiplications in 10+4\left(x-2\right).

$$\frac{-5+3x}{x-3}\left(\frac{10+4x-8}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

Combine like terms in 10+4x-8.

$$\frac{-5+3x}{x-3}\left(\frac{2+4x}{x\left(x-2\right)}-\frac{5}{x-2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right) and x-2 is x\left(x-2\right). Multiply \frac{5}{x-2} times \frac{x}{x}.

$$\frac{-5+3x}{x-3}\left(\frac{2+4x}{x\left(x-2\right)}-\frac{5x}{x\left(x-2\right)}\right)$$

Since \frac{2+4x}{x\left(x-2\right)} and \frac{5x}{x\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-5+3x}{x-3}\times \frac{2+4x-5x}{x\left(x-2\right)}$$

Combine like terms in 2+4x-5x.

$$\frac{-5+3x}{x-3}\times \frac{2-x}{x\left(x-2\right)}$$

Extract the negative sign in 2-x.

$$\frac{-5+3x}{x-3}\times \frac{-\left(x-2\right)}{x\left(x-2\right)}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{-5+3x}{x-3}\times \frac{-1}{x}$$

Multiply \frac{-5+3x}{x-3} times \frac{-1}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-5+3x\right)\left(-1\right)}{\left(x-3\right)x}$$

Use the distributive property to multiply -5+3x by -1.

$$\frac{5-3x}{\left(x-3\right)x}$$

Use the distributive property to multiply x-3 by x.

$$\frac{5-3x}{x^{2}-3x}$$