Algebra Examples

$| \frac{1}{3} - \frac{x}{5} | = \frac{7}{15}$

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$\frac{1}{3} - \frac{x}{5} = \pm \frac{7}{15}$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

$\frac{1}{3} - \frac{x}{5} = \frac{7}{15}$

Step 2.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 2.2.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$- \frac{x}{5} = \frac{7}{15} - \frac{1}{3}$

Step 2.2.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{x}{5} = \frac{7}{15} - \frac{1}{3} \cdot \frac{5}{5}$

Step 2.2.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.2.3.1

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$- \frac{x}{5} = \frac{7}{15} - \frac{5}{3 \cdot 5}$

Step 2.2.3.2

Multiply $3$ by $5$ .

$- \frac{x}{5} = \frac{7}{15} - \frac{5}{15}$

Step 2.2.4

Combine the numerators over the common denominator.

$- \frac{x}{5} = \frac{7 - 5}{15}$

Step 2.2.5

Subtract $5$ from $7$ .

$- \frac{x}{5} = \frac{2}{15}$

Step 2.3

Multiply both sides of the equation by $- 5$ .

$- 5 (- \frac{x}{5}) = - 5 (\frac{2}{15})$

Step 2.4

Simplify both sides of the equation.

Tap for more steps...

Step 2.4.1

Simplify the left side.

Tap for more steps...

Step 2.4.1.1

Simplify $- 5 (- \frac{x}{5})$ .

Tap for more steps...

Step 2.4.1.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.4.1.1.1.1

Move the leading negative in $- \frac{x}{5}$ into the numerator.

$- 5 \frac{- x}{5} = - 5 (\frac{2}{15})$

Step 2.4.1.1.1.2

Factor $5$ out of $- 5$ .

$5 (- 1) \frac{- x}{5} = - 5 (\frac{2}{15})$

Step 2.4.1.1.1.3

Cancel the common factor.

$5 \cdot - 1 \frac{- x}{5} = - 5 (\frac{2}{15})$

Step 2.4.1.1.1.4

Rewrite the expression.

$- - x = - 5 (\frac{2}{15})$

Step 2.4.1.1.2

Multiply.

Tap for more steps...

Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 5 (\frac{2}{15})$

Step 2.4.1.1.2.2

Multiply $x$ by $1$ .

$x = - 5 (\frac{2}{15})$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 2.4.2.1.1

Factor $5$ out of $- 5$ .

$x = 5 (- 1) \frac{2}{15}$

Step 2.4.2.1.2

Factor $5$ out of $15$ .

$x = 5 \cdot - 1 \frac{2}{5 \cdot 3}$

Step 2.4.2.1.3

Cancel the common factor.

$x = 5 \cdot - 1 \frac{2}{5 \cdot 3}$

Step 2.4.2.1.4

Rewrite the expression.

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

Step 2.5

Next, use the negative value of the $\pm$ to find the second solution.

$\frac{1}{3} - \frac{x}{5} = - \frac{7}{15}$

Step 2.6

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 2.6.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$- \frac{x}{5} = - \frac{7}{15} - \frac{1}{3}$

Step 2.6.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{x}{5} = - \frac{7}{15} - \frac{1}{3} \cdot \frac{5}{5}$

Step 2.6.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.6.3.1

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$- \frac{x}{5} = - \frac{7}{15} - \frac{5}{3 \cdot 5}$

Step 2.6.3.2

Multiply $3$ by $5$ .

$- \frac{x}{5} = - \frac{7}{15} - \frac{5}{15}$

Step 2.6.4

Combine the numerators over the common denominator.

$- \frac{x}{5} = \frac{- 7 - 5}{15}$

Step 2.6.5

Subtract $5$ from $- 7$ .

$- \frac{x}{5} = \frac{- 12}{15}$

Step 2.6.6

Cancel the common factor of $- 12$ and $15$ .

Tap for more steps...

Step 2.6.6.1

Factor $3$ out of $- 12$ .

$- \frac{x}{5} = \frac{3 (- 4)}{15}$

Step 2.6.6.2

Cancel the common factors.

Tap for more steps...

Step 2.6.6.2.1

Factor $3$ out of $15$ .

$- \frac{x}{5} = \frac{3 \cdot - 4}{3 \cdot 5}$

Step 2.6.6.2.2

Cancel the common factor.

$- \frac{x}{5} = \frac{3 \cdot - 4}{3 \cdot 5}$

Step 2.6.6.2.3

Rewrite the expression.

$- \frac{x}{5} = \frac{- 4}{5}$

Step 2.6.7

Move the negative in front of the fraction.

$- \frac{x}{5} = - \frac{4}{5}$

Step 2.7

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x = - 4$

Step 2.8

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

Tap for more steps...

Step 2.8.1

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

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

Step 2.8.2

Simplify the left side.

Tap for more steps...

Step 2.8.2.1

Dividing two negative values results in a positive value.

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

Step 2.8.2.2

Divide $x$ by $1$ .

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

Step 2.8.3

Simplify the right side.

Tap for more steps...

Step 2.8.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 2.9

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{2}{3}, 4$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{3}, 4$

Decimal Form:

$x = - 0.\overline{6}, 4$