Find Where Undefined/Discontinuous 5/(x-5)=6/x

$\frac{5}{x - 5} = \frac{6}{x}$

Step 1

Subtract $\frac{6}{x}$ from both sides of the equation.

$\frac{5}{x - 5} - \frac{6}{x} = 0$

Step 2

Simplify $\frac{5}{x - 5} - \frac{6}{x}$ .

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

$\frac{5}{x - 5} \cdot \frac{x}{x} - \frac{6}{x} = 0$

To write $- \frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{5}{x - 5} \cdot \frac{x}{x} - \frac{6}{x} \cdot \frac{x - 5}{x - 5} = 0$

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

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

$\frac{5 x}{(x - 5) x} - \frac{6}{x} \cdot \frac{x - 5}{x - 5} = 0$

Multiply $\frac{6}{x}$ by $\frac{x - 5}{x - 5}$ .

$\frac{5 x}{(x - 5) x} - \frac{6 (x - 5)}{x (x - 5)} = 0$

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

$\frac{5 x}{x (x - 5)} - \frac{6 (x - 5)}{x (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{5 x - 6 (x - 5)}{x (x - 5)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{5 x - 6 x - 6 \cdot - 5}{x (x - 5)} = 0$

Multiply $- 6$ by $- 5$ .

$\frac{5 x - 6 x + 30}{x (x - 5)} = 0$

Subtract $6 x$ from $5 x$ .

$\frac{- x + 30}{x (x - 5)} = 0$

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 30}{x (x - 5)} = 0$

Rewrite $30$ as $- 1 (- 30)$ .

$\frac{- (x) - 1 (- 30)}{x (x - 5)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 30)$ .

$\frac{- (x - 30)}{x (x - 5)} = 0$

Rewrite $- (x - 30)$ as $- 1 (x - 30)$ .

$\frac{- 1 (x - 30)}{x (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{x - 30}{x (x - 5)} = 0$

Step 3

Set the denominator in $\frac{x - 30}{x (x - 5)}$ equal to $0$ to find where the expression is undefined.

$x (x - 5) = 0$

Step 4

Solve for $x$ .

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

$x = 0$

$x - 5 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0, x = 5$

Step 6

Find Where Undefined/Discontinuous 5/(x-5)=6/x

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