Algebra Examples

$\frac{50}{x} - \frac{50}{x - 5} = - \frac{1}{2}$

Step 1

Add $\frac{1}{2}$ to both sides of the equation.

$\frac{50}{x} - \frac{50}{x - 5} + \frac{1}{2} = 0$

Step 2

Simplify $\frac{50}{x} - \frac{50}{x - 5} + \frac{1}{2}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{50}{x}$ by $\frac{(x - 5) \cdot 2}{(x - 5) \cdot 2}$ .

$\frac{50}{x} \cdot \frac{(x - 5) \cdot 2}{(x - 5) \cdot 2} - \frac{50}{x - 5} + \frac{1}{2} = 0$

Step 2.1.2

Multiply $\frac{50}{x}$ by $\frac{(x - 5) \cdot 2}{(x - 5) \cdot 2}$ .

$\frac{50 ((x - 5) \cdot 2)}{x ((x - 5) \cdot 2)} - \frac{50}{x - 5} + \frac{1}{2} = 0$

Step 2.1.3

Multiply $\frac{50}{x - 5}$ by $\frac{x \cdot 2}{x \cdot 2}$ .

$\frac{50 ((x - 5) \cdot 2)}{x ((x - 5) \cdot 2)} - (\frac{50}{x - 5} \cdot \frac{x \cdot 2}{x \cdot 2}) + \frac{1}{2} = 0$

Step 2.1.4

Multiply $\frac{50}{x - 5}$ by $\frac{x \cdot 2}{x \cdot 2}$ .

$\frac{50 ((x - 5) \cdot 2)}{x ((x - 5) \cdot 2)} - \frac{50 (x \cdot 2)}{(x - 5) (x \cdot 2)} + \frac{1}{2} = 0$

Step 2.1.5

Multiply $\frac{1}{2}$ by $\frac{x (x - 5)}{x (x - 5)}$ .

$\frac{50 ((x - 5) \cdot 2)}{x ((x - 5) \cdot 2)} - \frac{50 (x \cdot 2)}{(x - 5) (x \cdot 2)} + \frac{1}{2} \cdot \frac{x (x - 5)}{x (x - 5)} = 0$

Step 2.1.6

Multiply $\frac{1}{2}$ by $\frac{x (x - 5)}{x (x - 5)}$ .

$\frac{50 ((x - 5) \cdot 2)}{x ((x - 5) \cdot 2)} - \frac{50 (x \cdot 2)}{(x - 5) (x \cdot 2)} + \frac{x (x - 5)}{2 (x (x - 5))} = 0$

Step 2.1.7

Reorder the factors of $x ((x - 5) \cdot 2)$ .

$\frac{50 ((x - 5) \cdot 2)}{2 x (x - 5)} - \frac{50 (x \cdot 2)}{(x - 5) (x \cdot 2)} + \frac{x (x - 5)}{2 (x (x - 5))} = 0$

Step 2.1.8

Reorder the factors of $(x - 5) (x \cdot 2)$ .

$\frac{50 ((x - 5) \cdot 2)}{2 x (x - 5)} - \frac{50 (x \cdot 2)}{2 x (x - 5)} + \frac{x (x - 5)}{2 (x (x - 5))} = 0$

Step 2.1.9

Reorder the factors of $2 (x (x - 5))$ .

$\frac{50 ((x - 5) \cdot 2)}{2 x (x - 5)} - \frac{50 (x \cdot 2)}{2 x (x - 5)} + \frac{x (x - 5)}{2 x (x - 5)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{50 ((x - 5) \cdot 2) - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{50 (x \cdot 2 - 5 \cdot 2) - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.2

Move $2$ to the left of $x$ .

$\frac{50 (2 \cdot x - 5 \cdot 2) - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.3

Multiply $- 5$ by $2$ .

$\frac{50 (2 \cdot x - 10) - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{50 (2 x) + 50 \cdot - 10 - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.5

Multiply $2$ by $50$ .

$\frac{100 x + 50 \cdot - 10 - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.6

Multiply $50$ by $- 10$ .

$\frac{100 x - 500 - 50 (x \cdot 2) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.7

Move $2$ to the left of $x$ .

$\frac{100 x - 500 - 50 (2 \cdot x) + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.8

Multiply $2$ by $- 50$ .

$\frac{100 x - 500 - 100 x + x (x - 5)}{2 x (x - 5)} = 0$

Step 2.3.9

Apply the distributive property.

$\frac{100 x - 500 - 100 x + x \cdot x + x \cdot - 5}{2 x (x - 5)} = 0$

Step 2.3.10

Multiply $x$ by $x$ .

$\frac{100 x - 500 - 100 x + x^{2} + x \cdot - 5}{2 x (x - 5)} = 0$

Step 2.3.11

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

$\frac{100 x - 500 - 100 x + x^{2} - 5 x}{2 x (x - 5)} = 0$

Step 2.4

Combine the opposite terms in $100 x - 500 - 100 x + x^{2} - 5 x$ .

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Step 2.4.1

Subtract $100 x$ from $100 x$ .

$\frac{0 - 500 + x^{2} - 5 x}{2 x (x - 5)} = 0$

Step 2.4.2

Subtract $500$ from $0$ .

$\frac{- 500 + x^{2} - 5 x}{2 x (x - 5)} = 0$

Step 2.5

Factor $- 500 + x^{2} - 5 x$ 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 $- 500$ and whose sum is $- 5$ .

$- 25, 20$

Step 2.5.2

Write the factored form using these integers.

$\frac{(x - 25) (x + 20)}{2 x (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 25) (x + 20) = 0$

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 $0$ .

$x - 25 = 0$

$x + 20 = 0$

Step 4.2

Set $x - 25$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 25$ equal to $0$ .

$x - 25 = 0$

Step 4.2.2

Add $25$ to both sides of the equation.

$x = 25$

Step 4.3

Set $x + 20$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 20$ equal to $0$ .

$x + 20 = 0$

Step 4.3.2

Subtract $20$ from both sides of the equation.

$x = - 20$

Step 4.4

The final solution is all the values that make $(x - 25) (x + 20) = 0$ true.

$x = 25, - 20$