Solve (x-1)/(3x+3)-(x-2)/(6x-6)+((x)^(2)+2x-6)/(9(x)^(2)-9) | 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

$$\frac{ x-1 }{ 3x+3 } - \frac{ x-2 }{ 6x-6 } + \frac{ { x }^{ 2 } +2x-6 }{ 9 { x }^{ 2 } -9 }$$

Evaluate

$\frac{5x}{18\left(x+1\right)}$

Short Solution Steps

Factor $3x+3$. Factor $6x-6$.

$$\frac{x-1}{3\left(x+1\right)}-\frac{x-2}{6\left(x-1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3\left(x+1\right)$ and $6\left(x-1\right)$ is $6\left(x-1\right)\left(x+1\right)$. Multiply $\frac{x-1}{3\left(x+1\right)}$ times $\frac{2\left(x-1\right)}{2\left(x-1\right)}$. Multiply $\frac{x-2}{6\left(x-1\right)}$ times $\frac{x+1}{x+1}$.

$$\frac{\left(x-1\right)\times 2\left(x-1\right)}{6\left(x-1\right)\left(x+1\right)}-\frac{\left(x-2\right)\left(x+1\right)}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

Since $\frac{\left(x-1\right)\times 2\left(x-1\right)}{6\left(x-1\right)\left(x+1\right)}$ and $\frac{\left(x-2\right)\left(x+1\right)}{6\left(x-1\right)\left(x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-1\right)\times 2\left(x-1\right)-\left(x-2\right)\left(x+1\right)}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

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

$$\frac{2x^{2}-2x-2x+2-x^{2}-x+2x+2}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

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

$$\frac{x^{2}-3x+4}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

Factor $9x^{2}-9$.

$$\frac{x^{2}-3x+4}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9\left(x-1\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6\left(x-1\right)\left(x+1\right)$ and $9\left(x-1\right)\left(x+1\right)$ is $18\left(x-1\right)\left(x+1\right)$. Multiply $\frac{x^{2}-3x+4}{6\left(x-1\right)\left(x+1\right)}$ times $\frac{3}{3}$. Multiply $\frac{x^{2}+2x-6}{9\left(x-1\right)\left(x+1\right)}$ times $\frac{2}{2}$.

$$\frac{3\left(x^{2}-3x+4\right)}{18\left(x-1\right)\left(x+1\right)}+\frac{2\left(x^{2}+2x-6\right)}{18\left(x-1\right)\left(x+1\right)}$$

Since $\frac{3\left(x^{2}-3x+4\right)}{18\left(x-1\right)\left(x+1\right)}$ and $\frac{2\left(x^{2}+2x-6\right)}{18\left(x-1\right)\left(x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{3\left(x^{2}-3x+4\right)+2\left(x^{2}+2x-6\right)}{18\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in $3\left(x^{2}-3x+4\right)+2\left(x^{2}+2x-6\right)$.

$$\frac{3x^{2}-9x+12+2x^{2}+4x-12}{18\left(x-1\right)\left(x+1\right)}$$

Combine like terms in $3x^{2}-9x+12+2x^{2}+4x-12$.

$$\frac{5x^{2}-5x}{18\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in $\frac{5x^{2}-5x}{18\left(x-1\right)\left(x+1\right)}$.

$$\frac{5x\left(x-1\right)}{18\left(x-1\right)\left(x+1\right)}$$

Cancel out $x-1$ in both numerator and denominator.

$$\frac{5x}{18\left(x+1\right)}$$

Expand $18\left(x+1\right)$.

$$\frac{5x}{18x+18}$$

Expand

$\frac{5x}{18\left(x+1\right)}$

Short Solution Steps

Factor $3x+3$. Factor $6x-6$.

$$\frac{x-1}{3\left(x+1\right)}-\frac{x-2}{6\left(x-1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $3\left(x+1\right)$ and $6\left(x-1\right)$ is $6\left(x-1\right)\left(x+1\right)$. Multiply $\frac{x-1}{3\left(x+1\right)}$ times $\frac{2\left(x-1\right)}{2\left(x-1\right)}$. Multiply $\frac{x-2}{6\left(x-1\right)}$ times $\frac{x+1}{x+1}$.

$$\frac{\left(x-1\right)\times 2\left(x-1\right)}{6\left(x-1\right)\left(x+1\right)}-\frac{\left(x-2\right)\left(x+1\right)}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

Since $\frac{\left(x-1\right)\times 2\left(x-1\right)}{6\left(x-1\right)\left(x+1\right)}$ and $\frac{\left(x-2\right)\left(x+1\right)}{6\left(x-1\right)\left(x+1\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-1\right)\times 2\left(x-1\right)-\left(x-2\right)\left(x+1\right)}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

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

$$\frac{2x^{2}-2x-2x+2-x^{2}-x+2x+2}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

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

$$\frac{x^{2}-3x+4}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9x^{2}-9}$$

Factor $9x^{2}-9$.

$$\frac{x^{2}-3x+4}{6\left(x-1\right)\left(x+1\right)}+\frac{x^{2}+2x-6}{9\left(x-1\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $6\left(x-1\right)\left(x+1\right)$ and $9\left(x-1\right)\left(x+1\right)$ is $18\left(x-1\right)\left(x+1\right)$. Multiply $\frac{x^{2}-3x+4}{6\left(x-1\right)\left(x+1\right)}$ times $\frac{3}{3}$. Multiply $\frac{x^{2}+2x-6}{9\left(x-1\right)\left(x+1\right)}$ times $\frac{2}{2}$.

$$\frac{3\left(x^{2}-3x+4\right)}{18\left(x-1\right)\left(x+1\right)}+\frac{2\left(x^{2}+2x-6\right)}{18\left(x-1\right)\left(x+1\right)}$$

Since $\frac{3\left(x^{2}-3x+4\right)}{18\left(x-1\right)\left(x+1\right)}$ and $\frac{2\left(x^{2}+2x-6\right)}{18\left(x-1\right)\left(x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{3\left(x^{2}-3x+4\right)+2\left(x^{2}+2x-6\right)}{18\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in $3\left(x^{2}-3x+4\right)+2\left(x^{2}+2x-6\right)$.

$$\frac{3x^{2}-9x+12+2x^{2}+4x-12}{18\left(x-1\right)\left(x+1\right)}$$

Combine like terms in $3x^{2}-9x+12+2x^{2}+4x-12$.

$$\frac{5x^{2}-5x}{18\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in $\frac{5x^{2}-5x}{18\left(x-1\right)\left(x+1\right)}$.

$$\frac{5x\left(x-1\right)}{18\left(x-1\right)\left(x+1\right)}$$

Cancel out $x-1$ in both numerator and denominator.

$$\frac{5x}{18\left(x+1\right)}$$

Expand $18\left(x+1\right)$.

$$\frac{5x}{18x+18}$$