Solve the Absolute Value Inequality for x 2=|-4x+9|

$2 = | - 4 x + 9 |$

Step 1

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

$| - 4 x + 9 | = 2$

Step 2

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

$- 4 x + 9 = \pm 2$

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.

$- 4 x + 9 = 2$

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

Subtract $9$ from both sides of the equation.

$- 4 x = 2 - 9$

Subtract $9$ from $2$ .

$- 4 x = - 7$

Divide each term in $- 4 x = - 7$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 7$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 7}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 7}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{7}{4}$

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

$- 4 x + 9 = - 2$

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

Subtract $9$ from both sides of the equation.

$- 4 x = - 2 - 9$

Subtract $9$ from $- 2$ .

$- 4 x = - 11$

Divide each term in $- 4 x = - 11$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 11$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 11}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 11}{- 4}$

Divide $x$ by $1$ .

$x = \frac{- 11}{- 4}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{11}{4}$

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

$x = \frac{7}{4}, \frac{11}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{4}, \frac{11}{4}$

Decimal Form:

$x = 1.75, 2.75$

Mixed Number Form:

$x = 1 \frac{3}{4}, 2 \frac{3}{4}$

Solve the Absolute Value Inequality for x 2=|-4x+9|

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