Solve the Absolute Value Inequality for x 3-5|10-4x|=-57

$3 - 5 | 10 - 4 x | = - 57$

Step 1

Move all terms not containing $| 10 - 4 x |$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- 5 | 10 - 4 x | = - 57 - 3$

Subtract $3$ from $- 57$ .

$- 5 | 10 - 4 x | = - 60$

Step 2

Divide each term in $- 5 | 10 - 4 x | = - 60$ by $- 5$ and simplify.

Divide each term in $- 5 | 10 - 4 x | = - 60$ by $- 5$ .

$\frac{- 5 | 10 - 4 x |}{- 5} = \frac{- 60}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 | 10 - 4 x |}{- 5} = \frac{- 60}{- 5}$

Divide $| 10 - 4 x |$ by $1$ .

$| 10 - 4 x | = \frac{- 60}{- 5}$

Simplify the right side.

Divide $- 60$ by $- 5$ .

$| 10 - 4 x | = 12$

Step 3

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

$10 - 4 x = \pm 12$

Step 4

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.

$10 - 4 x = 12$

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

Subtract $10$ from both sides of the equation.

$- 4 x = 12 - 10$

Subtract $10$ from $12$ .

$- 4 x = 2$

Divide each term in $- 4 x = 2$ by $- 4$ and simplify.

Divide each term in $- 4 x = 2$ by $- 4$ .

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

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ and $- 4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x = \frac{1}{- 2}$

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

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

$10 - 4 x = - 12$

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

Subtract $10$ from both sides of the equation.

$- 4 x = - 12 - 10$

Subtract $10$ from $- 12$ .

$- 4 x = - 22$

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.

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

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

$x = - \frac{1}{2}, \frac{11}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, \frac{11}{2}$

Decimal Form:

$x = - 0.5, 5.5$

Mixed Number Form:

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

Solve the Absolute Value Inequality for x 3-5|10-4x|=-57

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