Algebra Examples

$\frac{4}{x} - \frac{1}{3} = \frac{x}{6}$

Step 1

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{4}{x} - \frac{1}{3} - \frac{x}{6} = 0$

Step 2

Reorder terms.

$- \frac{x}{6} + \frac{4}{x} - \frac{1}{3} = 0$

Step 3

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

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{4}{x} - \frac{1}{3} = 0$

Step 4

To write $\frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{4}{x} \cdot \frac{6}{6} - \frac{1}{3} = 0$

Step 5

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

Tap for more steps...

Step 5.1

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

$- \frac{x \cdot x}{6 x} + \frac{4}{x} \cdot \frac{6}{6} - \frac{1}{3} = 0$

Step 5.2

Multiply $\frac{4}{x}$ by $\frac{6}{6}$ .

$- \frac{x \cdot x}{6 x} + \frac{4 \cdot 6}{x \cdot 6} - \frac{1}{3} = 0$

Step 5.3

Reorder the factors of $x \cdot 6$ .

$- \frac{x \cdot x}{6 x} + \frac{4 \cdot 6}{6 x} - \frac{1}{3} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- x \cdot x + 4 \cdot 6}{6 x} - \frac{1}{3} = 0$

Step 7

Multiply $4$ by $6$ .

$\frac{- x \cdot x + 24}{6 x} - \frac{1}{3} = 0$

Step 8

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 8.1

Move $x$ .

$\frac{- (x \cdot x) + 24}{6 x} - \frac{1}{3} = 0$

Step 8.2

Multiply $x$ by $x$ .

$\frac{- x^{2} + 24}{6 x} - \frac{1}{3} = 0$

Step 9

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

$\frac{- x^{2} + 24}{6 x} - \frac{1}{3} \cdot \frac{2 x}{2 x} = 0$

Step 10

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

Tap for more steps...

Step 10.1

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

$\frac{- x^{2} + 24}{6 x} - \frac{2 x}{3 (2 x)} = 0$

Step 10.2

Multiply $2$ by $3$ .

$\frac{- x^{2} + 24}{6 x} - \frac{2 x}{6 x} = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{- x^{2} + 24 - 2 x}{6 x} = 0$

Step 12

Factor by grouping.

Tap for more steps...

Step 12.1

Reorder terms.

$\frac{- x^{2} - 2 x + 24}{6 x} = 0$

Step 12.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 24 = - 24$ and whose sum is $b = - 2$ .

Tap for more steps...

Step 12.2.1

Factor $- 2$ out of $- 2 x$ .

$\frac{- x^{2} - 2 x + 24}{6 x} = 0$

Step 12.2.2

Rewrite $- 2$ as $4$ plus $- 6$

$\frac{- x^{2} + (4 - 6) x + 24}{6 x} = 0$

Step 12.2.3

Apply the distributive property.

$\frac{- x^{2} + 4 x - 6 x + 24}{6 x} = 0$

Step 12.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 12.3.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} + 4 x) - 6 x + 24}{6 x} = 0$

Step 12.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x + 4) + 6 (- x + 4)}{6 x} = 0$

Step 12.4

Factor the polynomial by factoring out the greatest common factor, $- x + 4$ .

$\frac{(- x + 4) (x + 6)}{6 x} = 0$

Step 13

Set the numerator equal to zero.

$(- x + 4) (x + 6) = 0$

Step 14

Solve the equation for $x$ .

Tap for more steps...

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 $0$ .

$- x + 4 = 0$

$x + 6 = 0$

Step 14.2

Set $- x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 14.2.1

Set $- x + 4$ equal to $0$ .

$- x + 4 = 0$

Step 14.2.2

Solve $- x + 4 = 0$ for $x$ .

Tap for more steps...

Step 14.2.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 14.2.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 14.2.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 14.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 14.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 14.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 14.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 14.2.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 14.3

Set $x + 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 14.3.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 14.3.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 14.4

The final solution is all the values that make $(- x + 4) (x + 6) = 0$ true.

$x = 4, - 6$