Algebra Examples

$\frac{110}{x} - \frac{110}{x - 5} = - \frac{1}{5}$

Step 1

Add $\frac{1}{5}$ to both sides of the equation.

$\frac{110}{x} - \frac{110}{x - 5} + \frac{1}{5} = 0$

Step 2

Simplify $\frac{110}{x} - \frac{110}{x - 5} + \frac{1}{5}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{110}{x}$ by $\frac{(x - 5) \cdot 5}{(x - 5) \cdot 5}$ .

$\frac{110}{x} \cdot \frac{(x - 5) \cdot 5}{(x - 5) \cdot 5} - \frac{110}{x - 5} + \frac{1}{5} = 0$

Step 2.1.2

Multiply $\frac{110}{x}$ by $\frac{(x - 5) \cdot 5}{(x - 5) \cdot 5}$ .

$\frac{110 ((x - 5) \cdot 5)}{x ((x - 5) \cdot 5)} - \frac{110}{x - 5} + \frac{1}{5} = 0$

Step 2.1.3

Multiply $\frac{110}{x - 5}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

$\frac{110 ((x - 5) \cdot 5)}{x ((x - 5) \cdot 5)} - (\frac{110}{x - 5} \cdot \frac{x \cdot 5}{x \cdot 5}) + \frac{1}{5} = 0$

Step 2.1.4

Multiply $\frac{110}{x - 5}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

$\frac{110 ((x - 5) \cdot 5)}{x ((x - 5) \cdot 5)} - \frac{110 (x \cdot 5)}{(x - 5) (x \cdot 5)} + \frac{1}{5} = 0$

Step 2.1.5

Multiply $\frac{1}{5}$ by $\frac{x (x - 5)}{x (x - 5)}$ .

$\frac{110 ((x - 5) \cdot 5)}{x ((x - 5) \cdot 5)} - \frac{110 (x \cdot 5)}{(x - 5) (x \cdot 5)} + \frac{1}{5} \cdot \frac{x (x - 5)}{x (x - 5)} = 0$

Step 2.1.6

Multiply $\frac{1}{5}$ by $\frac{x (x - 5)}{x (x - 5)}$ .

$\frac{110 ((x - 5) \cdot 5)}{x ((x - 5) \cdot 5)} - \frac{110 (x \cdot 5)}{(x - 5) (x \cdot 5)} + \frac{x (x - 5)}{5 (x (x - 5))} = 0$

Step 2.1.7

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

$\frac{110 ((x - 5) \cdot 5)}{5 x (x - 5)} - \frac{110 (x \cdot 5)}{(x - 5) (x \cdot 5)} + \frac{x (x - 5)}{5 (x (x - 5))} = 0$

Step 2.1.8

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

$\frac{110 ((x - 5) \cdot 5)}{5 x (x - 5)} - \frac{110 (x \cdot 5)}{5 x (x - 5)} + \frac{x (x - 5)}{5 (x (x - 5))} = 0$

Step 2.1.9

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

$\frac{110 ((x - 5) \cdot 5)}{5 x (x - 5)} - \frac{110 (x \cdot 5)}{5 x (x - 5)} + \frac{x (x - 5)}{5 x (x - 5)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{110 ((x - 5) \cdot 5) - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{110 (x \cdot 5 - 5 \cdot 5) - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.2

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

$\frac{110 (5 \cdot x - 5 \cdot 5) - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.3

Multiply $- 5$ by $5$ .

$\frac{110 (5 \cdot x - 25) - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{110 (5 x) + 110 \cdot - 25 - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.5

Multiply $5$ by $110$ .

$\frac{550 x + 110 \cdot - 25 - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.6

Multiply $110$ by $- 25$ .

$\frac{550 x - 2750 - 110 (x \cdot 5) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.7

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

$\frac{550 x - 2750 - 110 (5 \cdot x) + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.8

Multiply $5$ by $- 110$ .

$\frac{550 x - 2750 - 550 x + x (x - 5)}{5 x (x - 5)} = 0$

Step 2.3.9

Apply the distributive property.

$\frac{550 x - 2750 - 550 x + x \cdot x + x \cdot - 5}{5 x (x - 5)} = 0$

Step 2.3.10

Multiply $x$ by $x$ .

$\frac{550 x - 2750 - 550 x + x^{2} + x \cdot - 5}{5 x (x - 5)} = 0$

Step 2.3.11

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

$\frac{550 x - 2750 - 550 x + x^{2} - 5 x}{5 x (x - 5)} = 0$

Step 2.4

Combine the opposite terms in $550 x - 2750 - 550 x + x^{2} - 5 x$ .

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Step 2.4.1

Subtract $550 x$ from $550 x$ .

$\frac{0 - 2750 + x^{2} - 5 x}{5 x (x - 5)} = 0$

Step 2.4.2

Subtract $2750$ from $0$ .

$\frac{- 2750 + x^{2} - 5 x}{5 x (x - 5)} = 0$

Step 2.5

Factor $- 2750 + x^{2} - 5 x$ using the AC method.

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Step 2.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2750$ and whose sum is $- 5$ .

$- 55, 50$

Step 2.5.2

Write the factored form using these integers.

$\frac{(x - 55) (x + 50)}{5 x (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 55) (x + 50) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

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

$x - 55 = 0$

$x + 50 = 0$

Step 4.2

Set $x - 55$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 55$ equal to $0$ .

$x - 55 = 0$

Step 4.2.2

Add $55$ to both sides of the equation.

$x = 55$

Step 4.3

Set $x + 50$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 50$ equal to $0$ .

$x + 50 = 0$

Step 4.3.2

Subtract $50$ from both sides of the equation.

$x = - 50$

Step 4.4

The final solution is all the values that make $(x - 55) (x + 50) = 0$ true.

$x = 55, - 50$