Solve the Rational Equation for x 1/(x^2-5x)=(x+7)/x-1

$\frac{1}{x^{2} - 5 x} = \frac{x + 7}{x} - 1$

Step 1

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

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

$\frac{1}{x \cdot x - 5 x} = \frac{x + 7}{x} - 1$

Factor $x$ out of $- 5 x$ .

$\frac{1}{x \cdot x + x \cdot - 5} = \frac{x + 7}{x} - 1$

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{1}{x (x - 5)} = \frac{x + 7}{x} - 1$

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x (x - 5), x, 1$

Since $x (x - 5), x, 1$ contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $x (x - 5), x, 1$ are:

1. Find the LCM for the numeric part $1, 1, 1$ .

2. Find the LCM for the variable part $x^{1}, x^{1}$ .

3. Find the LCM for the compound variable part $x - 5$ .

4. Multiply each LCM together.

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

The LCM of $x^{1}, x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

The factor for $x - 5$ is $x - 5$ itself.

$(x - 5) = x - 5$

$(x - 5)$ occurs $1$ time.

The LCM of $x - 5$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x - 5$

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$x (x - 5)$

Step 3

Multiply each term in $\frac{1}{x (x - 5)} = \frac{x + 7}{x} - 1$ by $x (x - 5)$ to eliminate the fractions.

Multiply each term in $\frac{1}{x (x - 5)} = \frac{x + 7}{x} - 1$ by $x (x - 5)$ .

$\frac{1}{x (x - 5)} (x (x - 5)) = \frac{x + 7}{x} (x (x - 5)) - (x (x - 5))$

Simplify the left side.

Cancel the common factor of $x (x - 5)$ .

Cancel the common factor.

$\frac{1}{x (x - 5)} (x (x - 5)) = \frac{x + 7}{x} (x (x - 5)) - (x (x - 5))$

Rewrite the expression.

$1 = \frac{x + 7}{x} (x (x - 5)) - (x (x - 5))$

Simplify the right side.

Simplify each term.

Cancel the common factor of $x$ .

Cancel the common factor.

$1 = \frac{x + 7}{x} (x (x - 5)) - (x (x - 5))$

Rewrite the expression.

$1 = (x + 7) (x - 5) - (x (x - 5))$

Expand $(x + 7) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$1 = x (x - 5) + 7 (x - 5) - (x (x - 5))$

Apply the distributive property.

$1 = x \cdot x + x \cdot - 5 + 7 (x - 5) - (x (x - 5))$

Apply the distributive property.

$1 = x \cdot x + x \cdot - 5 + 7 x + 7 \cdot - 5 - (x (x - 5))$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$1 = x^{2} + x \cdot - 5 + 7 x + 7 \cdot - 5 - (x (x - 5))$

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

$1 = x^{2} - 5 \cdot x + 7 x + 7 \cdot - 5 - (x (x - 5))$

Multiply $7$ by $- 5$ .

$1 = x^{2} - 5 x + 7 x - 35 - (x (x - 5))$

Add $- 5 x$ and $7 x$ .

$1 = x^{2} + 2 x - 35 - (x (x - 5))$

Apply the distributive property.

$1 = x^{2} + 2 x - 35 - (x \cdot x + x \cdot - 5)$

Multiply $x$ by $x$ .

$1 = x^{2} + 2 x - 35 - (x^{2} + x \cdot - 5)$

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

$1 = x^{2} + 2 x - 35 - (x^{2} - 5 \cdot x)$

Apply the distributive property.

$1 = x^{2} + 2 x - 35 - x^{2} - (- 5 x)$

Multiply $- 5$ by $- 1$ .

$1 = x^{2} + 2 x - 35 - x^{2} + 5 x$

Simplify by adding terms.

Combine the opposite terms in $x^{2} + 2 x - 35 - x^{2} + 5 x$ .

Subtract $x^{2}$ from $x^{2}$ .

$1 = 2 x - 35 + 0 + 5 x$

Add $2 x - 35$ and $0$ .

$1 = 2 x - 35 + 5 x$

Add $2 x$ and $5 x$ .

$1 = 7 x - 35$

Step 4

Solve the equation.

Rewrite the equation as $7 x - 35 = 1$ .

$7 x - 35 = 1$

Move all terms not containing $x$ to the right side of the equation.

Add $35$ to both sides of the equation.

$7 x = 1 + 35$

Add $1$ and $35$ .

$7 x = 36$

Divide each term in $7 x = 36$ by $7$ and simplify.

Divide each term in $7 x = 36$ by $7$ .

$\frac{7 x}{7} = \frac{36}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{36}{7}$

Divide $x$ by $1$ .

$x = \frac{36}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{36}{7}$

Decimal Form:

$x = 5.\overline{142857}$

Mixed Number Form:

$x = 5 \frac{1}{7}$

Solve the Rational Equation for x 1/(x^2-5x)=(x+7)/x-1

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