Algebra Examples

$\frac{x}{3} - \frac{1}{x} = \frac{2}{3}$

Step 1

Subtract $\frac{2}{3}$ from both sides of the equation.

$\frac{x}{3} - \frac{1}{x} - \frac{2}{3} = 0$

Step 2

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

$\frac{x}{3} \cdot \frac{x}{x} - \frac{1}{x} - \frac{2}{3} = 0$

Step 3

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

$\frac{x}{3} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{3}{3} - \frac{2}{3} = 0$

Step 4

Write each expression with a common denominator of $3 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

$\frac{x \cdot x}{3 x} - \frac{1}{x} \cdot \frac{3}{3} - \frac{2}{3} = 0$

Step 4.2

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

$\frac{x \cdot x}{3 x} - \frac{3}{x \cdot 3} - \frac{2}{3} = 0$

Step 4.3

Reorder the factors of $x \cdot 3$ .

$\frac{x \cdot x}{3 x} - \frac{3}{3 x} - \frac{2}{3} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{x \cdot x - 3}{3 x} - \frac{2}{3} = 0$

Step 6

Multiply $x$ by $x$ .

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

Step 7

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

$\frac{x^{2} - 3}{3 x} - \frac{2}{3} \cdot \frac{x}{x} = 0$

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

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

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Step 10.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 10.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 1)}{3 x} = 0$

Step 11

Set the numerator equal to zero.

$(x - 3) (x + 1) = 0$

Step 12

Solve the equation for $x$ .

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Step 12.1

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

$x - 3 = 0$

$x + 1 = 0$

Step 12.2

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 12.2.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 12.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 12.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 12.3.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 12.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 12.4

The final solution is all the values that make $(x - 3) (x + 1) = 0$ true.

$x = 3, - 1$