Solve x+24/x-2+frac{1{2-x}}

$$\frac { x + 24 } { x - 2 + \frac { 1 } { 2 - x } }$$

Evaluate

$$\frac{\left(2-x\right)\left(x+24\right)}{\left(1-x\right)\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x-2 times \frac{2-x}{2-x}.

$$\frac{x+24}{\frac{\left(x-2\right)\left(2-x\right)}{2-x}+\frac{1}{2-x}}$$

Since \frac{\left(x-2\right)\left(2-x\right)}{2-x} and \frac{1}{2-x} have the same denominator, add them by adding their numerators.

$$\frac{x+24}{\frac{\left(x-2\right)\left(2-x\right)+1}{2-x}}$$

Do the multiplications in \left(x-2\right)\left(2-x\right)+1.

$$\frac{x+24}{\frac{2x-x^{2}-4+2x+1}{2-x}}$$

Combine like terms in 2x-x^{2}-4+2x+1.

$$\frac{x+24}{\frac{4x-x^{2}-3}{2-x}}$$

Divide x+24 by \frac{4x-x^{2}-3}{2-x} by multiplying x+24 by the reciprocal of \frac{4x-x^{2}-3}{2-x}.

$$\frac{\left(x+24\right)\left(2-x\right)}{4x-x^{2}-3}$$

Apply the distributive property by multiplying each term of x+24 by each term of 2-x.

$$\frac{2x-x^{2}+48-24x}{4x-x^{2}-3}$$

Combine 2x and -24x to get -22x.

$$\frac{-22x-x^{2}+48}{4x-x^{2}-3}$$

Expand

$$\frac{48-22x-x^{2}}{-x^{2}+4x-3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x-2 times \frac{2-x}{2-x}.

$$\frac{x+24}{\frac{\left(x-2\right)\left(2-x\right)}{2-x}+\frac{1}{2-x}}$$

Since \frac{\left(x-2\right)\left(2-x\right)}{2-x} and \frac{1}{2-x} have the same denominator, add them by adding their numerators.

$$\frac{x+24}{\frac{\left(x-2\right)\left(2-x\right)+1}{2-x}}$$

Do the multiplications in \left(x-2\right)\left(2-x\right)+1.

$$\frac{x+24}{\frac{2x-x^{2}-4+2x+1}{2-x}}$$

Combine like terms in 2x-x^{2}-4+2x+1.

$$\frac{x+24}{\frac{4x-x^{2}-3}{2-x}}$$

Divide x+24 by \frac{4x-x^{2}-3}{2-x} by multiplying x+24 by the reciprocal of \frac{4x-x^{2}-3}{2-x}.

$$\frac{\left(x+24\right)\left(2-x\right)}{4x-x^{2}-3}$$

Apply the distributive property by multiplying each term of x+24 by each term of 2-x.

$$\frac{2x-x^{2}+48-24x}{4x-x^{2}-3}$$

Combine 2x and -24x to get -22x.

$$\frac{-22x-x^{2}+48}{4x-x^{2}-3}$$