Calculus Examples

$2 | 4 - \frac{3}{2} x | + 9 = 25$

Step 1

Move all terms not containing $| 4 - \frac{3}{2} x |$ to the right side of the equation.

Tap for more steps...

Step 1.1

Subtract $9$ from both sides of the equation.

$2 | 4 - \frac{3}{2} x | = 25 - 9$

Step 1.2

Subtract $9$ from $25$ .

$2 | 4 - \frac{3}{2} x | = 16$

Step 2

Divide each term in $2 | 4 - \frac{3}{2} x | = 16$ by $2$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $2 | 4 - \frac{3}{2} x | = 16$ by $2$ .

$\frac{2 | 4 - \frac{3}{2} x |}{2} = \frac{16}{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{2 | 4 - \frac{3}{2} x |}{2} = \frac{16}{2}$

Step 2.2.1.2

Divide $| 4 - \frac{3}{2} x |$ by $1$ .

$| 4 - \frac{3}{2} x | = \frac{16}{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Divide $16$ by $2$ .

$| 4 - \frac{3}{2} x | = 8$

Step 3

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

$4 - \frac{3}{2} x = \pm 8$

Step 4

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

Tap for more steps...

Step 4.1

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

$4 - \frac{3}{2} x = 8$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Combine $x$ and $\frac{3}{2}$ .

$4 - \frac{x \cdot 3}{2} = 8$

Step 4.2.2

Move $3$ to the left of $x$ .

$4 - \frac{3 x}{2} = 8$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Subtract $4$ from both sides of the equation.

$- \frac{3 x}{2} = 8 - 4$

Step 4.3.2

Subtract $4$ from $8$ .

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

Step 4.4

Multiply both sides of the equation by $- \frac{2}{3}$ .

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

Step 4.5

Simplify both sides of the equation.

Tap for more steps...

Step 4.5.1

Simplify the left side.

Tap for more steps...

Step 4.5.1.1

Simplify $- \frac{2}{3} (- \frac{3 x}{2})$ .

Tap for more steps...

Step 4.5.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.5.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Step 4.5.1.1.1.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

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

Step 4.5.1.1.1.3

Factor $2$ out of $- 2$ .

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

Step 4.5.1.1.1.4

Cancel the common factor.

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

Step 4.5.1.1.1.5

Rewrite the expression.

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

Step 4.5.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.5.1.1.2.1

Factor $3$ out of $- 3 x$ .

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

Step 4.5.1.1.2.2

Cancel the common factor.

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

Step 4.5.1.1.2.3

Rewrite the expression.

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

Step 4.5.1.1.3

Multiply.

Tap for more steps...

Step 4.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.1.1.3.2

Multiply $x$ by $1$ .

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

Step 4.5.2

Simplify the right side.

Tap for more steps...

Step 4.5.2.1

Simplify $- \frac{2}{3} \cdot 4$ .

Tap for more steps...

Step 4.5.2.1.1

Multiply $- \frac{2}{3} \cdot 4$ .

Tap for more steps...

Step 4.5.2.1.1.1

Multiply $4$ by $- 1$ .

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

Step 4.5.2.1.1.2

Combine $- 4$ and $\frac{2}{3}$ .

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

Step 4.5.2.1.1.3

Multiply $- 4$ by $2$ .

$x = \frac{- 8}{3}$

Step 4.5.2.1.2

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 4.6

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

$4 - \frac{3}{2} x = - 8$

Step 4.7

Simplify each term.

Tap for more steps...

Step 4.7.1

Combine $x$ and $\frac{3}{2}$ .

$4 - \frac{x \cdot 3}{2} = - 8$

Step 4.7.2

Move $3$ to the left of $x$ .

$4 - \frac{3 x}{2} = - 8$

Step 4.8

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

Tap for more steps...

Step 4.8.1

Subtract $4$ from both sides of the equation.

$- \frac{3 x}{2} = - 8 - 4$

Step 4.8.2

Subtract $4$ from $- 8$ .

$- \frac{3 x}{2} = - 12$

Step 4.9

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot - 12$

Step 4.10

Simplify both sides of the equation.

Tap for more steps...

Step 4.10.1

Simplify the left side.

Tap for more steps...

Step 4.10.1.1

Simplify $- \frac{2}{3} (- \frac{3 x}{2})$ .

Tap for more steps...

Step 4.10.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.10.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x}{2}) = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.1.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x}{2} = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x) = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.10.1.1.2.1

Factor $3$ out of $- 3 x$ .

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x)) = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.2.3

Rewrite the expression.

$- - x = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.3

Multiply.

Tap for more steps...

Step 4.10.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{2}{3} \cdot - 12$

Step 4.10.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{2}{3} \cdot - 12$

Step 4.10.2

Simplify the right side.

Tap for more steps...

Step 4.10.2.1

Simplify $- \frac{2}{3} \cdot - 12$ .

Tap for more steps...

Step 4.10.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.10.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x = \frac{- 2}{3} \cdot - 12$

Step 4.10.2.1.1.2

Factor $3$ out of $- 12$ .

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

Step 4.10.2.1.1.3

Cancel the common factor.

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

Step 4.10.2.1.1.4

Rewrite the expression.

$x = - 2 \cdot - 4$

Step 4.10.2.1.2

Multiply $- 2$ by $- 4$ .

$x = 8$

Step 4.11

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

$x = - \frac{8}{3}, 8$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{8}{3}, 8$

Decimal Form:

$x = - 2.\overline{6}, 8$

Mixed Number Form:

$x = - 2 \frac{2}{3}, 8$