Solve (x)/((x)^(2)+2x-3)+(x-3)/((1-x)(x+2))-(1)/(x+2) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$\frac{x^{2}-12}{\left(1-x\right)\left(x+2\right)\left(x+3\right)}$

Short Solution Steps

Factor $x^{2}+2x-3$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-1\right)\left(x+3\right)$ and $\left(1-x\right)\left(x+2\right)$ is $\left(x-1\right)\left(x+2\right)\left(x+3\right)$. Multiply $\frac{x}{\left(x-1\right)\left(x+3\right)}$ times $\frac{x+2}{x+2}$. Multiply $\frac{x-3}{\left(1-x\right)\left(x+2\right)}$ times $\frac{-\left(x+3\right)}{-\left(x+3\right)}$.

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

Since $\frac{x\left(x+2\right)}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ and $\frac{\left(x-3\right)\left(-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $x\left(x+2\right)+\left(x-3\right)\left(-1\right)\left(x+3\right)$.

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

Combine like terms in $x^{2}+2x-x^{2}-3x+3x+9$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-1\right)\left(x+2\right)\left(x+3\right)$ and $x+2$ is $\left(x-1\right)\left(x+2\right)\left(x+3\right)$. Multiply $\frac{1}{x+2}$ times $\frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}$.

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

Since $\frac{2x+9}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ and $\frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in $2x+9-\left(x-1\right)\left(x+3\right)$.

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

Combine like terms in $2x+9-x^{2}-3x+x+3$.

$$\frac{12-x^{2}}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Expand $\left(x-1\right)\left(x+2\right)\left(x+3\right)$.

$$\frac{12-x^{2}}{x^{3}+4x^{2}+x-6}$$

Expand

$-\frac{x^{2}-12}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$

Short Solution Steps

Factor $x^{2}+2x-3$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-1\right)\left(x+3\right)$ and $\left(1-x\right)\left(x+2\right)$ is $\left(x-1\right)\left(x+2\right)\left(x+3\right)$. Multiply $\frac{x}{\left(x-1\right)\left(x+3\right)}$ times $\frac{x+2}{x+2}$. Multiply $\frac{x-3}{\left(1-x\right)\left(x+2\right)}$ times $\frac{-\left(x+3\right)}{-\left(x+3\right)}$.

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

Since $\frac{x\left(x+2\right)}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ and $\frac{\left(x-3\right)\left(-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $x\left(x+2\right)+\left(x-3\right)\left(-1\right)\left(x+3\right)$.

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

Combine like terms in $x^{2}+2x-x^{2}-3x+3x+9$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-1\right)\left(x+2\right)\left(x+3\right)$ and $x+2$ is $\left(x-1\right)\left(x+2\right)\left(x+3\right)$. Multiply $\frac{1}{x+2}$ times $\frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}$.

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

Since $\frac{2x+9}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ and $\frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$ have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in $2x+9-\left(x-1\right)\left(x+3\right)$.

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

Combine like terms in $2x+9-x^{2}-3x+x+3$.

$$\frac{12-x^{2}}{\left(x-1\right)\left(x+2\right)\left(x+3\right)}$$

Expand $\left(x-1\right)\left(x+2\right)\left(x+3\right)$.

$$\frac{12-x^{2}}{x^{3}+4x^{2}+x-6}$$