Find All Complex Solutions |x-1|=5x+2

$| x - 1 | = 5 x + 2$

Step 1

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

$x - 1 = \pm (5 x + 2)$

Step 2

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.

$x - 1 = 5 x + 2$

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

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

$x - 1 - 5 x = 2$

Subtract $5 x$ from $x$ .

$- 4 x - 1 = 2$

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

Add $1$ to both sides of the equation.

$- 4 x = 2 + 1$

Add $2$ and $1$ .

$- 4 x = 3$

Divide each term in $- 4 x = 3$ by $- 4$ and simplify.

Divide each term in $- 4 x = 3$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{3}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{3}{- 4}$

Divide $x$ by $1$ .

$x = \frac{3}{- 4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

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

$x - 1 = - (5 x + 2)$

Simplify $- (5 x + 2)$ .

Rewrite.

$x - 1 = 0 + 0 - (5 x + 2)$

Simplify by adding zeros.

$x - 1 = - (5 x + 2)$

Apply the distributive property.

$x - 1 = - (5 x) - 1 \cdot 2$

Multiply.

Multiply $5$ by $- 1$ .

$x - 1 = - 5 x - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$x - 1 = - 5 x - 2$

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

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

$x - 1 + 5 x = - 2$

Add $x$ and $5 x$ .

$6 x - 1 = - 2$

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

Add $1$ to both sides of the equation.

$6 x = - 2 + 1$

Add $- 2$ and $1$ .

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

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

$x = - \frac{3}{4}, - \frac{1}{6}$

Step 3

Exclude the solutions that do not make $| x - 1 | = 5 x + 2$ true.

$x = - \frac{1}{6}$

Find All Complex Solutions |x-1|=5x+2

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