Precalculus Examples

$| 2 x - 1 | = | 4 x + 9 |$

Step 1

Rewrite the equation as $| 4 x + 9 | = | 2 x - 1 |$ .

$| 4 x + 9 | = | 2 x - 1 |$

Step 2

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

$4 x + 9 = 2 x - 1$

$4 x + 9 = - (2 x - 1)$

$- (4 x + 9) = 2 x - 1$

$- (4 x + 9) = - (2 x - 1)$

Step 3

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

$4 x + 9 = 2 x - 1$

$4 x + 9 = - (2 x - 1)$

Step 4

Solve $4 x + 9 = 2 x - 1$ for $x$ .

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Subtract $2 x$ from both sides of the equation.

$4 x + 9 - 2 x = - 1$

Step 4.1.2

Subtract $2 x$ from $4 x$ .

$2 x + 9 = - 1$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $9$ from both sides of the equation.

$2 x = - 1 - 9$

Step 4.2.2

Subtract $9$ from $- 1$ .

$2 x = - 10$

Step 4.3

Divide each term in $2 x = - 10$ by $2$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $2 x = - 10$ by $2$ .

$\frac{2 x}{2} = \frac{- 10}{2}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 10}{2}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{2}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Divide $- 10$ by $2$ .

$x = - 5$

Step 5

Solve $4 x + 9 = - (2 x - 1)$ for $x$ .

Tap for more steps...

Step 5.1

Simplify $- (2 x - 1)$ .

Tap for more steps...

Step 5.1.1

Rewrite.

$4 x + 9 = 0 + 0 - (2 x - 1)$

Step 5.1.2

Simplify by adding zeros.

$4 x + 9 = - (2 x - 1)$

Step 5.1.3

Apply the distributive property.

$4 x + 9 = - (2 x) - - 1$

Step 5.1.4

Multiply.

Tap for more steps...

Step 5.1.4.1

Multiply $2$ by $- 1$ .

$4 x + 9 = - 2 x - - 1$

Step 5.1.4.2

Multiply $- 1$ by $- 1$ .

$4 x + 9 = - 2 x + 1$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Add $2 x$ to both sides of the equation.

$4 x + 9 + 2 x = 1$

Step 5.2.2

Add $4 x$ and $2 x$ .

$6 x + 9 = 1$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Subtract $9$ from both sides of the equation.

$6 x = 1 - 9$

Step 5.3.2

Subtract $9$ from $1$ .

$6 x = - 8$

Step 5.4

Divide each term in $6 x = - 8$ by $6$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $6 x = - 8$ by $6$ .

$\frac{6 x}{6} = \frac{- 8}{6}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 8}{6}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{6}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Cancel the common factor of $- 8$ and $6$ .

Tap for more steps...

Step 5.4.3.1.1

Factor $2$ out of $- 8$ .

$x = \frac{2 (- 4)}{6}$

Step 5.4.3.1.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.2.1

Factor $2$ out of $6$ .

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

Step 5.4.3.1.2.2

Cancel the common factor.

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

Step 5.4.3.1.2.3

Rewrite the expression.

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

Step 5.4.3.2

Move the negative in front of the fraction.

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

Step 6

List all of the solutions.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 5, - 1.\overline{3}$

Mixed Number Form:

$x = - 5, - 1 \frac{1}{3}$