Find the Excluded Values 7/x-9/(x-4)=5

$\frac{7}{x} - \frac{9}{x - 4} = 5$

Step 1

Subtract $5$ from both sides of the equation.

$\frac{7}{x} - \frac{9}{x - 4} - 5 = 0$

Step 2

Simplify $\frac{7}{x} - \frac{9}{x - 4} - 5$ .

To write $\frac{7}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{7}{x} \cdot \frac{x - 4}{x - 4} - \frac{9}{x - 4} - 5 = 0$

To write $- \frac{9}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{7}{x} \cdot \frac{x - 4}{x - 4} - \frac{9}{x - 4} \cdot \frac{x}{x} - 5 = 0$

Write each expression with a common denominator of $x (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{x}$ by $\frac{x - 4}{x - 4}$ .

$\frac{7 (x - 4)}{x (x - 4)} - \frac{9}{x - 4} \cdot \frac{x}{x} - 5 = 0$

Multiply $\frac{9}{x - 4}$ by $\frac{x}{x}$ .

$\frac{7 (x - 4)}{x (x - 4)} - \frac{9 x}{(x - 4) x} - 5 = 0$

Reorder the factors of $(x - 4) x$ .

$\frac{7 (x - 4)}{x (x - 4)} - \frac{9 x}{x (x - 4)} - 5 = 0$

Combine the numerators over the common denominator.

$\frac{7 (x - 4) - 9 x}{x (x - 4)} - 5 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{7 x + 7 \cdot - 4 - 9 x}{x (x - 4)} - 5 = 0$

Multiply $7$ by $- 4$ .

$\frac{7 x - 28 - 9 x}{x (x - 4)} - 5 = 0$

Subtract $9 x$ from $7 x$ .

$\frac{- 2 x - 28}{x (x - 4)} - 5 = 0$

Factor $2$ out of $- 2 x - 28$ .

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) - 28}{x (x - 4)} - 5 = 0$

Factor $2$ out of $- 28$ .

$\frac{2 (- x) + 2 (- 14)}{x (x - 4)} - 5 = 0$

Factor $2$ out of $2 (- x) + 2 (- 14)$ .

$\frac{2 (- x - 14)}{x (x - 4)} - 5 = 0$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x (x - 4)}{x (x - 4)}$ .

$\frac{2 (- x - 14)}{x (x - 4)} - 5 \cdot \frac{x (x - 4)}{x (x - 4)} = 0$

Combine $- 5$ and $\frac{x (x - 4)}{x (x - 4)}$ .

$\frac{2 (- x - 14)}{x (x - 4)} + \frac{- 5 (x (x - 4))}{x (x - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (- x - 14) - 5 (x (x - 4))}{x (x - 4)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2 (- x) + 2 \cdot - 14 - 5 x (x - 4)}{x (x - 4)} = 0$

Multiply $- 1$ by $2$ .

$\frac{- 2 x + 2 \cdot - 14 - 5 x (x - 4)}{x (x - 4)} = 0$

Multiply $2$ by $- 14$ .

$\frac{- 2 x - 28 - 5 x (x - 4)}{x (x - 4)} = 0$

Apply the distributive property.

$\frac{- 2 x - 28 - 5 x \cdot x - 5 x \cdot - 4}{x (x - 4)} = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{- 2 x - 28 - 5 (x \cdot x) - 5 x \cdot - 4}{x (x - 4)} = 0$

Multiply $x$ by $x$ .

$\frac{- 2 x - 28 - 5 x^{2} - 5 x \cdot - 4}{x (x - 4)} = 0$

Multiply $- 4$ by $- 5$ .

$\frac{- 2 x - 28 - 5 x^{2} + 20 x}{x (x - 4)} = 0$

Add $- 2 x$ and $20 x$ .

$\frac{18 x - 28 - 5 x^{2}}{x (x - 4)} = 0$

Reorder terms.

$\frac{- 5 x^{2} + 18 x - 28}{x (x - 4)} = 0$

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) + 18 x - 28}{x (x - 4)} = 0$

Factor $- 1$ out of $18 x$ .

$\frac{- (5 x^{2}) - (- 18 x) - 28}{x (x - 4)} = 0$

Factor $- 1$ out of $- (5 x^{2}) - (- 18 x)$ .

$\frac{- (5 x^{2} - 18 x) - 28}{x (x - 4)} = 0$

Rewrite $- 28$ as $- 1 (28)$ .

$\frac{- (5 x^{2} - 18 x) - 1 (28)}{x (x - 4)} = 0$

Factor $- 1$ out of $- (5 x^{2} - 18 x) - 1 (28)$ .

$\frac{- (5 x^{2} - 18 x + 28)}{x (x - 4)} = 0$

Rewrite $- (5 x^{2} - 18 x + 28)$ as $- 1 (5 x^{2} - 18 x + 28)$ .

$\frac{- 1 (5 x^{2} - 18 x + 28)}{x (x - 4)} = 0$

Move the negative in front of the fraction.

$- \frac{5 x^{2} - 18 x + 28}{x (x - 4)} = 0$

Step 3

Set the denominator in $\frac{5 x^{2} - 18 x + 28}{x (x - 4)}$ equal to $0$ to find where the expression is undefined.

$x (x - 4) = 0$

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 $0$ .

$x = 0$

$x - 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

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 $0$ .

$x = 0, x = 4$

Step 6

Find the Excluded Values 7/x-9/(x-4)=5

Related Questions

Support LCMGCF.com by disabling your adblocker.