Find All Complex Solutions -|x-4|+2=-2x+7

$- | x - 4 | + 2 = - 2 x + 7$

Step 1

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

Subtract $2$ from both sides of the equation.

$- | x - 4 | = - 2 x + 7 - 2$

Subtract $2$ from $7$ .

$- | x - 4 | = - 2 x + 5$

Step 2

Divide each term in $- | x - 4 | = - 2 x + 5$ by $- 1$ and simplify.

Divide each term in $- | x - 4 | = - 2 x + 5$ by $- 1$ .

$\frac{- | x - 4 |}{- 1} = \frac{- 2 x}{- 1} + \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{| x - 4 |}{1} = \frac{- 2 x}{- 1} + \frac{5}{- 1}$

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

$| x - 4 | = \frac{- 2 x}{- 1} + \frac{5}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$| x - 4 | = - 1 \cdot (- 2 x) + \frac{5}{- 1}$

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$| x - 4 | = - (- 2 x) + \frac{5}{- 1}$

Multiply $- 2$ by $- 1$ .

$| x - 4 | = 2 x + \frac{5}{- 1}$

Divide $5$ by $- 1$ .

$| x - 4 | = 2 x - 5$

Step 3

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

$x - 4 = \pm (2 x - 5)$

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.

$x - 4 = 2 x - 5$

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

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

$x - 4 - 2 x = - 5$

Subtract $2 x$ from $x$ .

$- x - 4 = - 5$

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

Add $4$ to both sides of the equation.

$- x = - 5 + 4$

Add $- 5$ and $4$ .

$- x = - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

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

$x - 4 = - (2 x - 5)$

Simplify $- (2 x - 5)$ .

Rewrite.

$x - 4 = 0 + 0 - (2 x - 5)$

Simplify by adding zeros.

$x - 4 = - (2 x - 5)$

Apply the distributive property.

$x - 4 = - (2 x) - - 5$

Multiply.

Multiply $2$ by $- 1$ .

$x - 4 = - 2 x - - 5$

Multiply $- 1$ by $- 5$ .

$x - 4 = - 2 x + 5$

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

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

$x - 4 + 2 x = 5$

Add $x$ and $2 x$ .

$3 x - 4 = 5$

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

Add $4$ to both sides of the equation.

$3 x = 5 + 4$

Add $5$ and $4$ .

$3 x = 9$

Divide each term in $3 x = 9$ by $3$ and simplify.

Divide each term in $3 x = 9$ by $3$ .

$\frac{3 x}{3} = \frac{9}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{9}{3}$

Divide $x$ by $1$ .

$x = \frac{9}{3}$

Simplify the right side.

Divide $9$ by $3$ .

$x = 3$

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

$x = 1, 3$

Step 5

Exclude the solutions that do not make $- | x - 4 | + 2 = - 2 x + 7$ true.

$x = 3$

Find All Complex Solutions -|x-4|+2=-2x+7

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