Lutasin ang (x-2/x+1+5-x/x-2):[(1/x^2-x-2-1/x^2+3x+2)*(frac{x+1}{x}+frac{3-x^2}{x^2+x})]

$$( \frac { x - 2 } { x + 1 } + \frac { 5 - x } { x - 2 } ) : [ ( \frac { 1 } { x ^ { 2 } - x - 2 } - \frac { 1 } { x ^ { 2 } + 3 x + 2 } ) \cdot ( \frac { x + 1 } { x } + \frac { 3 - x ^ { 2 } } { x ^ { 2 } + x } ) ]$$

I-evaluate

$$\frac{9x\left(x+1\right)}{8}$$

Maiikling Hakbang sa Solution

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng x+1 at x-2 ay \left(x-2\right)\left(x+1\right). I-multiply ang \frac{x-2}{x+1} times \frac{x-2}{x-2}. I-multiply ang \frac{5-x}{x-2} times \frac{x+1}{x+1}.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Dahil may parehong denominator ang \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} at \frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Gawin ang mga pag-multiply sa \left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right).

$$\frac{\frac{x^{2}-2x-2x+4+5x+5-x^{2}-x}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Pagsamahin ang magkakatulad na term sa x^{2}-2x-2x+4+5x+5-x^{2}-x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

I-factor out ang x^{2}-x-2. I-factor out ang x^{2}+3x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{\left(x-2\right)\left(x+1\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng \left(x-2\right)\left(x+1\right) at \left(x+1\right)\left(x+2\right) ay \left(x-2\right)\left(x+1\right)\left(x+2\right). I-multiply ang \frac{1}{\left(x-2\right)\left(x+1\right)} times \frac{x+2}{x+2}. I-multiply ang \frac{1}{\left(x+1\right)\left(x+2\right)} times \frac{x-2}{x-2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Dahil may parehong denominator ang \frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} at \frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}, ibawas ang mga ito sa pamamagitan ng pagbawas sa mga numerator ng mga ito.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Gawin ang mga pag-multiply sa x+2-\left(x-2\right).

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Pagsamahin ang magkakatulad na term sa x+2-x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

I-factor out ang x^{2}+x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng x at x\left(x+1\right) ay x\left(x+1\right). I-multiply ang \frac{x+1}{x} times \frac{x+1}{x+1}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

Dahil may parehong denominator ang \frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)} at \frac{3-x^{2}}{x\left(x+1\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{\left(x+1\right)\left(x+1\right)+3-x^{2}}{x\left(x+1\right)}}$$

Gawin ang mga pag-multiply sa \left(x+1\right)\left(x+1\right)+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{x^{2}+x+1+x+3-x^{2}}{x\left(x+1\right)}}$$

Pagsamahin ang magkakatulad na term sa x^{2}+x+1+x+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{2x+4}{x\left(x+1\right)}}$$

I-multiply ang \frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} sa \frac{2x+4}{x\left(x+1\right)} sa pamamagitan ng pag-multiply ng numerator sa numerator at denominator sa denominator.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}}$$

I-divide ang \frac{9}{\left(x-2\right)\left(x+1\right)} gamit ang \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)} sa pamamagitan ng pagmu-multiply sa \frac{9}{\left(x-2\right)\left(x+1\right)} gamit ang reciprocal ng \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}.

$$\frac{9\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\times 4\left(2x+4\right)}$$

I-cancel out ang \left(x-2\right)\left(x+1\right) sa parehong numerator at denominator.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{4\left(2x+4\right)}$$

I-factor ang mga expression na hindi pa na-factor.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{2\times 4\left(x+2\right)}$$

I-cancel out ang x+2 sa parehong numerator at denominator.

$$\frac{9x\left(x+1\right)}{2\times 4}$$

Palawakin ang expression.

$$\frac{9x^{2}+9x}{8}$$

Palawakin

$$\frac{9x^{2}+9x}{8}$$

Maiikling Hakbang sa Solution

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng x+1 at x-2 ay \left(x-2\right)\left(x+1\right). I-multiply ang \frac{x-2}{x+1} times \frac{x-2}{x-2}. I-multiply ang \frac{5-x}{x-2} times \frac{x+1}{x+1}.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Dahil may parehong denominator ang \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} at \frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Gawin ang mga pag-multiply sa \left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right).

$$\frac{\frac{x^{2}-2x-2x+4+5x+5-x^{2}-x}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Pagsamahin ang magkakatulad na term sa x^{2}-2x-2x+4+5x+5-x^{2}-x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

I-factor out ang x^{2}-x-2. I-factor out ang x^{2}+3x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{\left(x-2\right)\left(x+1\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng \left(x-2\right)\left(x+1\right) at \left(x+1\right)\left(x+2\right) ay \left(x-2\right)\left(x+1\right)\left(x+2\right). I-multiply ang \frac{1}{\left(x-2\right)\left(x+1\right)} times \frac{x+2}{x+2}. I-multiply ang \frac{1}{\left(x+1\right)\left(x+2\right)} times \frac{x-2}{x-2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Dahil may parehong denominator ang \frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} at \frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}, ibawas ang mga ito sa pamamagitan ng pagbawas sa mga numerator ng mga ito.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Gawin ang mga pag-multiply sa x+2-\left(x-2\right).

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Pagsamahin ang magkakatulad na term sa x+2-x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

I-factor out ang x^{2}+x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

Para magdagdag o mag-subtract ng mga expression, i-expand ang mga iyon para gawing magkakapareho ang mga denominator ng mga ito. Ang least common multiple ng x at x\left(x+1\right) ay x\left(x+1\right). I-multiply ang \frac{x+1}{x} times \frac{x+1}{x+1}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

Dahil may parehong denominator ang \frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)} at \frac{3-x^{2}}{x\left(x+1\right)}, pagsamahin ang mga ito sa pamamagitan ng pagsasama sa mga numerator ng mga ito.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{\left(x+1\right)\left(x+1\right)+3-x^{2}}{x\left(x+1\right)}}$$

Gawin ang mga pag-multiply sa \left(x+1\right)\left(x+1\right)+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{x^{2}+x+1+x+3-x^{2}}{x\left(x+1\right)}}$$

Pagsamahin ang magkakatulad na term sa x^{2}+x+1+x+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{2x+4}{x\left(x+1\right)}}$$

I-multiply ang \frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} sa \frac{2x+4}{x\left(x+1\right)} sa pamamagitan ng pag-multiply ng numerator sa numerator at denominator sa denominator.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}}$$

I-divide ang \frac{9}{\left(x-2\right)\left(x+1\right)} gamit ang \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)} sa pamamagitan ng pagmu-multiply sa \frac{9}{\left(x-2\right)\left(x+1\right)} gamit ang reciprocal ng \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}.

$$\frac{9\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\times 4\left(2x+4\right)}$$

I-cancel out ang \left(x-2\right)\left(x+1\right) sa parehong numerator at denominator.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{4\left(2x+4\right)}$$

I-factor ang mga expression na hindi pa na-factor.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{2\times 4\left(x+2\right)}$$

I-cancel out ang x+2 sa parehong numerator at denominator.

$$\frac{9x\left(x+1\right)}{2\times 4}$$

Palawakin ang expression.

$$\frac{9x^{2}+9x}{8}$$