Solve for z |5z-2|=|z+6|

$| 5 z - 2 | = | z + 6 |$

Step 1

Rewrite the equation as $| z + 6 | = | 5 z - 2 |$ .

$| z + 6 | = | 5 z - 2 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$z + 6 = 5 z - 2$

$z + 6 = - (5 z - 2)$

$- (z + 6) = 5 z - 2$

$- (z + 6) = - (5 z - 2)$

Step 3

After simplifying, there are only two unique equations to be solved.

$z + 6 = 5 z - 2$

$z + 6 = - (5 z - 2)$

Step 4

Solve $z + 6 = 5 z - 2$ for $z$ .

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

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

$z + 6 - 5 z = - 2$

Subtract $5 z$ from $z$ .

$- 4 z + 6 = - 2$

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

Subtract $6$ from both sides of the equation.

$- 4 z = - 2 - 6$

Subtract $6$ from $- 2$ .

$- 4 z = - 8$

Divide each term in $- 4 z = - 8$ by $- 4$ and simplify.

Divide each term in $- 4 z = - 8$ by $- 4$ .

$\frac{- 4 z}{- 4} = \frac{- 8}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 z}{- 4} = \frac{- 8}{- 4}$

Divide $z$ by $1$ .

$z = \frac{- 8}{- 4}$

Simplify the right side.

Divide $- 8$ by $- 4$ .

$z = 2$

Step 5

Solve $z + 6 = - (5 z - 2)$ for $z$ .

Simplify $- (5 z - 2)$ .

Rewrite.

$z + 6 = 0 + 0 - (5 z - 2)$

Simplify by adding zeros.

$z + 6 = - (5 z - 2)$

Apply the distributive property.

$z + 6 = - (5 z) - - 2$

Multiply.

Multiply $5$ by $- 1$ .

$z + 6 = - 5 z - - 2$

Multiply $- 1$ by $- 2$ .

$z + 6 = - 5 z + 2$

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

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

$z + 6 + 5 z = 2$

Add $z$ and $5 z$ .

$6 z + 6 = 2$

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

Subtract $6$ from both sides of the equation.

$6 z = 2 - 6$

Subtract $6$ from $2$ .

$6 z = - 4$

Divide each term in $6 z = - 4$ by $6$ and simplify.

Divide each term in $6 z = - 4$ by $6$ .

$\frac{6 z}{6} = \frac{- 4}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 z}{6} = \frac{- 4}{6}$

Divide $z$ by $1$ .

$z = \frac{- 4}{6}$

Simplify the right side.

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $- 4$ .

$z = \frac{2 (- 2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$z = \frac{2 \cdot - 2}{2 \cdot 3}$

Cancel the common factor.

$z = \frac{2 \cdot - 2}{2 \cdot 3}$

Rewrite the expression.

$z = \frac{- 2}{3}$

Move the negative in front of the fraction.

$z = - \frac{2}{3}$

Step 6

List all of the solutions.

$z = 2, - \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = 2, - \frac{2}{3}$

Decimal Form:

$z = 2, - 0.\overline{6}$

Solve for z |5z-2|=|z+6|

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