Algebra Examples

$| \frac{1}{2} n + 2 | = | \frac{3}{4} n - 2 |$

Step 1

Rewrite the equation as $| \frac{3}{4} n - 2 | = | \frac{1}{2} n + 2 |$ .

$| \frac{3}{4} n - 2 | = | \frac{1}{2} n + 2 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$\frac{3}{4} n - 2 = \frac{1}{2} n + 2$

$\frac{3}{4} n - 2 = - (\frac{1}{2} n + 2)$

$- (\frac{3}{4} n - 2) = \frac{1}{2} n + 2$

$- (\frac{3}{4} n - 2) = - (\frac{1}{2} n + 2)$

Step 3

After simplifying, there are only two unique equations to be solved.

$\frac{3}{4} n - 2 = \frac{1}{2} n + 2$

$\frac{3}{4} n - 2 = - (\frac{1}{2} n + 2)$

Step 4

Solve $\frac{3}{4} n - 2 = \frac{1}{2} n + 2$ for $n$ .

Tap for more steps...

Step 4.1

Combine $\frac{3}{4}$ and $n$ .

$\frac{3 n}{4} - 2 = \frac{1}{2} n + 2$

Step 4.2

Combine $\frac{1}{2}$ and $n$ .

$\frac{3 n}{4} - 2 = \frac{n}{2} + 2$

Step 4.3

Move all terms containing $n$ to the left side of the equation.

Tap for more steps...

Step 4.3.1

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

$\frac{3 n}{4} - 2 - \frac{n}{2} = 2$

Step 4.3.2

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

$\frac{3 n}{4} - \frac{n}{2} \cdot \frac{2}{2} - 2 = 2$

Step 4.3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.3.3.1

Multiply $\frac{n}{2}$ by $\frac{2}{2}$ .

$\frac{3 n}{4} - \frac{n \cdot 2}{2 \cdot 2} - 2 = 2$

Step 4.3.3.2

Multiply $2$ by $2$ .

$\frac{3 n}{4} - \frac{n \cdot 2}{4} - 2 = 2$

Step 4.3.4

Combine the numerators over the common denominator.

$\frac{3 n - n \cdot 2}{4} - 2 = 2$

Step 4.3.5

Simplify each term.

Tap for more steps...

Step 4.3.5.1

Simplify the numerator.

Tap for more steps...

Step 4.3.5.1.1

Factor $n$ out of $3 n - n \cdot 2$ .

Tap for more steps...

Step 4.3.5.1.1.1

Factor $n$ out of $3 n$ .

$\frac{n \cdot 3 - n \cdot 2}{4} - 2 = 2$

Step 4.3.5.1.1.2

Factor $n$ out of $- n \cdot 2$ .

$\frac{n \cdot 3 + n (- 1 \cdot 2)}{4} - 2 = 2$

Step 4.3.5.1.1.3

Factor $n$ out of $n \cdot 3 + n (- 1 \cdot 2)$ .

$\frac{n (3 - 1 \cdot 2)}{4} - 2 = 2$

Step 4.3.5.1.2

Multiply $- 1$ by $2$ .

$\frac{n (3 - 2)}{4} - 2 = 2$

Step 4.3.5.1.3

Subtract $2$ from $3$ .

$\frac{n \cdot 1}{4} - 2 = 2$

Step 4.3.5.2

Multiply $n$ by $1$ .

$\frac{n}{4} - 2 = 2$

Step 4.4

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

Tap for more steps...

Step 4.4.1

Add $2$ to both sides of the equation.

$\frac{n}{4} = 2 + 2$

Step 4.4.2

Add $2$ and $2$ .

$\frac{n}{4} = 4$

Step 4.5

Multiply both sides of the equation by $4$ .

$4 \frac{n}{4} = 4 \cdot 4$

Step 4.6

Simplify both sides of the equation.

Tap for more steps...

Step 4.6.1

Simplify the left side.

Tap for more steps...

Step 4.6.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.6.1.1.1

Cancel the common factor.

$4 \frac{n}{4} = 4 \cdot 4$

Step 4.6.1.1.2

Rewrite the expression.

$n = 4 \cdot 4$

Step 4.6.2

Simplify the right side.

Tap for more steps...

Step 4.6.2.1

Multiply $4$ by $4$ .

$n = 16$

Step 5

Solve $\frac{3}{4} n - 2 = - (\frac{1}{2} n + 2)$ for $n$ .

Tap for more steps...

Step 5.1

Combine $\frac{3}{4}$ and $n$ .

$\frac{3 n}{4} - 2 = - (\frac{1}{2} n + 2)$

Step 5.2

Simplify $- (\frac{1}{2} n + 2)$ .

Tap for more steps...

Step 5.2.1

Combine $\frac{1}{2}$ and $n$ .

$\frac{3 n}{4} - 2 = - (\frac{n}{2} + 2)$

Step 5.2.2

Apply the distributive property.

$\frac{3 n}{4} - 2 = - \frac{n}{2} - 1 \cdot 2$

Step 5.2.3

Multiply $- 1$ by $2$ .

$\frac{3 n}{4} - 2 = - \frac{n}{2} - 2$

Step 5.3

Move all terms containing $n$ to the left side of the equation.

Tap for more steps...

Step 5.3.1

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

$\frac{3 n}{4} - 2 + \frac{n}{2} = - 2$

Step 5.3.2

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

$\frac{3 n}{4} + \frac{n}{2} \cdot \frac{2}{2} - 2 = - 2$

Step 5.3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.3.3.1

Multiply $\frac{n}{2}$ by $\frac{2}{2}$ .

$\frac{3 n}{4} + \frac{n \cdot 2}{2 \cdot 2} - 2 = - 2$

Step 5.3.3.2

Multiply $2$ by $2$ .

$\frac{3 n}{4} + \frac{n \cdot 2}{4} - 2 = - 2$

Step 5.3.4

Combine the numerators over the common denominator.

$\frac{3 n + n \cdot 2}{4} - 2 = - 2$

Step 5.3.5

Add $3 n$ and $n \cdot 2$ .

Tap for more steps...

Step 5.3.5.1

Reorder $n$ and $2$ .

$\frac{3 n + 2 \cdot n}{4} - 2 = - 2$

Step 5.3.5.2

Add $3 n$ and $2 \cdot n$ .

$\frac{5 n}{4} - 2 = - 2$

Step 5.4

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

Tap for more steps...

Step 5.4.1

Add $2$ to both sides of the equation.

$\frac{5 n}{4} = - 2 + 2$

Step 5.4.2

Add $- 2$ and $2$ .

$\frac{5 n}{4} = 0$

Step 5.5

Set the numerator equal to zero.

$5 n = 0$

Step 5.6

Divide each term in $5 n = 0$ by $5$ and simplify.

Tap for more steps...

Step 5.6.1

Divide each term in $5 n = 0$ by $5$ .

$\frac{5 n}{5} = \frac{0}{5}$

Step 5.6.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.6.2.1.1

Cancel the common factor.

$\frac{5 n}{5} = \frac{0}{5}$

Step 5.6.2.1.2

Divide $n$ by $1$ .

$n = \frac{0}{5}$

Step 5.6.3

Simplify the right side.

Tap for more steps...

Step 5.6.3.1

Divide $0$ by $5$ .

$n = 0$

Step 6

List all of the solutions.

$n = 16, 0$