Solve by Factoring (x+3)/(x-3)+x/(x-5)=(x+5)/(x-5)

$\frac{x + 3}{x - 3} + \frac{x}{x - 5} = \frac{x + 5}{x - 5}$

Step 1

Subtract $\frac{x + 5}{x - 5}$ from both sides of the equation.

$\frac{x + 3}{x - 3} + \frac{x}{x - 5} - \frac{x + 5}{x - 5} = 0$

Step 2

Simplify $\frac{x + 3}{x - 3} + \frac{x}{x - 5} - \frac{x + 5}{x - 5}$ .

Combine the numerators over the common denominator.

$\frac{x + 3}{x - 3} + \frac{x - (x + 5)}{x - 5} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x + 3}{x - 3} + \frac{x - x - 1 \cdot 5}{x - 5} = 0$

Multiply $- 1$ by $5$ .

$\frac{x + 3}{x - 3} + \frac{x - x - 5}{x - 5} = 0$

Subtract $x$ from $x$ .

$\frac{x + 3}{x - 3} + \frac{0 - 5}{x - 5} = 0$

Subtract $5$ from $0$ .

$\frac{x + 3}{x - 3} + \frac{- 5}{x - 5} = 0$

Move the negative in front of the fraction.

$\frac{x + 3}{x - 3} - \frac{5}{x - 5} = 0$

To write $\frac{x + 3}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{x + 3}{x - 3} \cdot \frac{x - 5}{x - 5} - \frac{5}{x - 5} = 0$

To write $- \frac{5}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 3}{x - 3} \cdot \frac{x - 5}{x - 5} - \frac{5}{x - 5} \cdot \frac{x - 3}{x - 3} = 0$

Write each expression with a common denominator of $(x - 3) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 3}{x - 3}$ by $\frac{x - 5}{x - 5}$ .

$\frac{(x + 3) (x - 5)}{(x - 3) (x - 5)} - \frac{5}{x - 5} \cdot \frac{x - 3}{x - 3} = 0$

Multiply $\frac{5}{x - 5}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x + 3) (x - 5)}{(x - 3) (x - 5)} - \frac{5 (x - 3)}{(x - 5) (x - 3)} = 0$

Reorder the factors of $(x - 5) (x - 3)$ .

$\frac{(x + 3) (x - 5)}{(x - 3) (x - 5)} - \frac{5 (x - 3)}{(x - 3) (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 3) (x - 5) - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Simplify the numerator.

Expand $(x + 3) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x - 5) + 3 (x - 5) - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 + 3 (x - 5) - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 + 3 x + 3 \cdot - 5 - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 5 + 3 x + 3 \cdot - 5 - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Move $- 5$ to the left of $x$ .

$\frac{x^{2} - 5 \cdot x + 3 x + 3 \cdot - 5 - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Multiply $3$ by $- 5$ .

$\frac{x^{2} - 5 x + 3 x - 15 - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Add $- 5 x$ and $3 x$ .

$\frac{x^{2} - 2 x - 15 - 5 (x - 3)}{(x - 3) (x - 5)} = 0$

Apply the distributive property.

$\frac{x^{2} - 2 x - 15 - 5 x - 5 \cdot - 3}{(x - 3) (x - 5)} = 0$

Multiply $- 5$ by $- 3$ .

$\frac{x^{2} - 2 x - 15 - 5 x + 15}{(x - 3) (x - 5)} = 0$

Subtract $5 x$ from $- 2 x$ .

$\frac{x^{2} - 7 x - 15 + 15}{(x - 3) (x - 5)} = 0$

Add $- 15$ and $15$ .

$\frac{x^{2} - 7 x + 0}{(x - 3) (x - 5)} = 0$

Add $x^{2} - 7 x$ and $0$ .

$\frac{x^{2} - 7 x}{(x - 3) (x - 5)} = 0$

Factor $x$ out of $x^{2} - 7 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x - 7 x}{(x - 3) (x - 5)} = 0$

Factor $x$ out of $- 7 x$ .

$\frac{x \cdot x + x \cdot - 7}{(x - 3) (x - 5)} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 7$ .

$\frac{x (x - 7)}{(x - 3) (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$x (x - 7) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 7 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $x (x - 7) = 0$ true.

$x = 0, 7$

Solve by Factoring (x+3)/(x-3)+x/(x-5)=(x+5)/(x-5)

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