Algebra Examples

$\frac{1}{2} + \frac{x}{6} = \frac{18}{x}$

Step 1

Subtract $\frac{18}{x}$ from both sides of the equation.

$\frac{1}{2} + \frac{x}{6} - \frac{18}{x} = 0$

Step 2

Reorder terms.

$\frac{x}{6} - \frac{18}{x} + \frac{1}{2} = 0$

Step 3

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

$\frac{x}{6} \cdot \frac{x}{x} - \frac{18}{x} + \frac{1}{2} = 0$

Step 4

To write $- \frac{18}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{6} \cdot \frac{x}{x} - \frac{18}{x} \cdot \frac{6}{6} + \frac{1}{2} = 0$

Step 5

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

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

$\frac{x \cdot x}{6 x} - \frac{18}{x} \cdot \frac{6}{6} + \frac{1}{2} = 0$

Step 5.2

Multiply $\frac{18}{x}$ by $\frac{6}{6}$ .

$\frac{x \cdot x}{6 x} - \frac{18 \cdot 6}{x \cdot 6} + \frac{1}{2} = 0$

Step 5.3

Reorder the factors of $x \cdot 6$ .

$\frac{x \cdot x}{6 x} - \frac{18 \cdot 6}{6 x} + \frac{1}{2} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{x \cdot x - 18 \cdot 6}{6 x} + \frac{1}{2} = 0$

Step 7

Multiply $- 18$ by $6$ .

$\frac{x \cdot x - 108}{6 x} + \frac{1}{2} = 0$

Step 8

Multiply $x$ by $x$ .

$\frac{x^{2} - 108}{6 x} + \frac{1}{2} = 0$

Step 9

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3 x}{3 x}$ .

$\frac{x^{2} - 108}{6 x} + \frac{1}{2} \cdot \frac{3 x}{3 x} = 0$

Step 10

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{1}{2}$ by $\frac{3 x}{3 x}$ .

$\frac{x^{2} - 108}{6 x} + \frac{3 x}{2 (3 x)} = 0$

Step 10.2

Multiply $3$ by $2$ .

$\frac{x^{2} - 108}{6 x} + \frac{3 x}{6 x} = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{x^{2} - 108 + 3 x}{6 x} = 0$

Step 12

Factor $x^{2} - 108 + 3 x$ using the AC method.

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Step 12.1

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 $- 108$ and whose sum is $3$ .

$- 9, 12$

Step 12.2

Write the factored form using these integers.

$\frac{(x - 9) (x + 12)}{6 x} = 0$

Step 13

Set the numerator equal to zero.

$(x - 9) (x + 12) = 0$

Step 14

Solve the equation for $x$ .

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Step 14.1

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

$x - 9 = 0$

$x + 12 = 0$

Step 14.2

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 14.2.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 14.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 14.3

Set $x + 12$ equal to $0$ and solve for $x$ .

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Step 14.3.1

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Step 14.3.2

Subtract $12$ from both sides of the equation.

$x = - 12$

Step 14.4

The final solution is all the values that make $(x - 9) (x + 12) = 0$ true.

$x = 9, - 12$