Solve the Absolute Value Inequality for z z-10=|7z|-16

$z - 10 = | 7 z | - 16$

Rewrite the equation as $| 7 z | - 16 = z - 10$ .

$| 7 z | - 16 = z - 10$

Move all terms not containing $| 7 z |$ to the right side of the equation.

Add $16$ to both sides of the equation.

$| 7 z | = z - 10 + 16$

Add $- 10$ and $16$ .

$| 7 z | = z + 6$

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

$7 z = \pm (z + 6)$

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.

$7 z = z + 6$

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

Subtract $z$ from both sides of the equation.

$7 z - z = 6$

Subtract $z$ from $7 z$ .

$6 z = 6$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $z$ by $1$ .

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

Simplify the right side.

Divide $6$ by $6$ .

$z = 1$

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

$7 z = - (z + 6)$

Simplify $- (z + 6)$ .

Apply the distributive property.

$7 z = - z - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$7 z = - z - 6$

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

Add $z$ to both sides of the equation.

$7 z + z = - 6$

Add $7 z$ and $z$ .

$8 z = - 6$

Divide each term in $8 z = - 6$ by $8$ and simplify.

Divide each term in $8 z = - 6$ by $8$ .

$\frac{8 z}{8} = \frac{- 6}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 z}{8} = \frac{- 6}{8}$

Divide $z$ by $1$ .

$z = \frac{- 6}{8}$

Simplify the right side.

Cancel the common factor of $- 6$ and $8$ .

Factor $2$ out of $- 6$ .

$z = \frac{2 (- 3)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$z = \frac{2 \cdot - 3}{2 \cdot 4}$

Cancel the common factor.

$z = \frac{2 \cdot - 3}{2 \cdot 4}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

$z = 1, - \frac{3}{4}$

The result can be shown in multiple forms.

Exact Form:

$z = 1, - \frac{3}{4}$

Decimal Form:

$z = 1, - 0.75$

Solve the Absolute Value Inequality for z z-10=|7z|-16

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