Solve the Absolute Value Inequality for x |4x-9|=5x

$| 4 x - 9 | = 5 x$

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

$4 x - 9 = \pm 5 x$

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.

$4 x - 9 = 5 x$

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

Subtract $5 x$ from both sides of the equation.

$4 x - 9 - 5 x = 0$

Subtract $5 x$ from $4 x$ .

$- x - 9 = 0$

Add $9$ to both sides of the equation.

$- x = 9$

Divide each term in $- x = 9$ by $- 1$ and simplify.

Divide each term in $- x = 9$ by $- 1$ .

$\frac{- x}{- 1} = \frac{9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{9}{- 1}$

Divide $x$ by $1$ .

$x = \frac{9}{- 1}$

Simplify the right side.

Divide $9$ by $- 1$ .

$x = - 9$

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

$4 x - 9 = - 5 x$

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

Add $5 x$ to both sides of the equation.

$4 x - 9 + 5 x = 0$

Add $4 x$ and $5 x$ .

$9 x - 9 = 0$

Add $9$ to both sides of the equation.

$9 x = 9$

Divide each term in $9 x = 9$ by $9$ and simplify.

Divide each term in $9 x = 9$ by $9$ .

$\frac{9 x}{9} = \frac{9}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{9}{9}$

Divide $x$ by $1$ .

$x = \frac{9}{9}$

Simplify the right side.

Divide $9$ by $9$ .

$x = 1$

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

$x = - 9, 1$

Exclude the solutions that do not make $| 4 x - 9 | = 5 x$ true.

$x = 1$

Solve the Absolute Value Inequality for x |4x-9|=5x

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