Solve the Absolute Value Inequality for x 11=|11+4x|

$11 = | 11 + 4 x |$

Step 1

Rewrite the equation as $| 11 + 4 x | = 11$ .

$| 11 + 4 x | = 11$

Step 2

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

$11 + 4 x = \pm 11$

Step 3

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

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

$11 + 4 x = 11$

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

Subtract $11$ from both sides of the equation.

$4 x = 11 - 11$

Subtract $11$ from $11$ .

$4 x = 0$

Divide each term in $4 x = 0$ by $4$ and simplify.

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x = 0$

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

$11 + 4 x = - 11$

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

Subtract $11$ from both sides of the equation.

$4 x = - 11 - 11$

Subtract $11$ from $- 11$ .

$4 x = - 22$

Divide each term in $4 x = - 22$ by $4$ and simplify.

Divide each term in $4 x = - 22$ by $4$ .

$\frac{4 x}{4} = \frac{- 22}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 22}{4}$

Divide $x$ by $1$ .

$x = \frac{- 22}{4}$

Simplify the right side.

Cancel the common factor of $- 22$ and $4$ .

Factor $2$ out of $- 22$ .

$x = \frac{2 (- 11)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 11}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 \cdot - 11}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{- 11}{2}$

Move the negative in front of the fraction.

$x = - \frac{11}{2}$

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

$x = 0, - \frac{11}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{11}{2}$

Decimal Form:

$x = 0, - 5.5$

Mixed Number Form:

$x = 0, - 5 \frac{1}{2}$

Solve the Absolute Value Inequality for x 11=|11+4x|

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