Simplify |2z-5|+2|10-4z|=15

$| 2 z - 5 | + 2 | 10 - 4 z | = 15$

Subtract $2 | 10 - 4 z |$ from both sides of the equation.

$| 2 z - 5 | = 15 - 2 | 10 - 4 z |$

Reorder $15$ and $- 2 | 10 - 4 z |$ .

$| 2 z - 5 | = - 2 | 10 - 4 z | + 15$

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

$2 z - 5 = \pm (- 2 | 10 - 4 z | + 15)$

The result consists of both the positive and negative portions of the $\pm$ .

$2 z - 5 = - 2 | 10 - 4 z | + 15$

$2 z - 5 = - (- 2 | 10 - 4 z | + 15)$

Solve $2 z - 5 = - 2 | 10 - 4 z | + 15$ for $z$ .

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Solve for $| 10 - 4 z |$ .

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Rewrite the equation as $- 2 | 10 - 4 z | + 15 = 2 z - 5$ .

$- 2 | 10 - 4 z | + 15 = 2 z - 5$

Move all terms not containing $| 10 - 4 z |$ to the right side of the equation.

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Subtract $15$ from both sides of the equation.

$- 2 | 10 - 4 z | = 2 z - 5 - 15$

Subtract $15$ from $- 5$ .

$- 2 | 10 - 4 z | = 2 z - 20$

Divide each term by $- 2$ and simplify.

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Divide each term in $- 2 | 10 - 4 z | = 2 z - 20$ by $- 2$ .

$\frac{- 2 | 10 - 4 z |}{- 2} = \frac{2 z}{- 2} + \frac{- 20}{- 2}$

Cancel the common factor of $- 2$ .

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Cancel the common factor.

$\frac{- 2 | 10 - 4 z |}{- 2} = \frac{2 z}{- 2} + \frac{- 20}{- 2}$

Divide $| 10 - 4 z |$ by $1$ .

$| 10 - 4 z | = \frac{2 z}{- 2} + \frac{- 20}{- 2}$

Simplify each term.

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Cancel the common factor of $2$ and $- 2$ .

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Factor $2$ out of $2 z$ .

$| 10 - 4 z | = \frac{2 (z)}{- 2} + \frac{- 20}{- 2}$

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

$| 10 - 4 z | = - 1 \cdot z + \frac{- 20}{- 2}$

Rewrite $- 1 \cdot z$ as $- z$ .

$| 10 - 4 z | = - z + \frac{- 20}{- 2}$

Divide $- 20$ by $- 2$ .

$| 10 - 4 z | = - z + 10$

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

$10 - 4 z = \pm (- z + 10)$

The result consists of both the positive and negative portions of the $\pm$ .

$10 - 4 z = - z + 10$

$10 - 4 z = - (- z + 10)$

Solve $10 - 4 z = - z + 10$ for $z$ .

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

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Add $z$ to both sides of the equation.

$10 - 4 z + z = 10$

Add $- 4 z$ and $z$ .

$10 - 3 z = 10$

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

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Subtract $10$ from both sides of the equation.

$- 3 z = 10 - 10$

Subtract $10$ from $10$ .

$- 3 z = 0$

Divide each term by $- 3$ and simplify.

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Divide each term in $- 3 z = 0$ by $- 3$ .

$\frac{- 3 z}{- 3} = \frac{0}{- 3}$

Cancel the common factor of $- 3$ .

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Cancel the common factor.

$\frac{- 3 z}{- 3} = \frac{0}{- 3}$

Divide $z$ by $1$ .

$z = \frac{0}{- 3}$

Divide $0$ by $- 3$ .

$z = 0$

Solve $10 - 4 z = - (- z + 10)$ for $z$ .

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Simplify $- (- z + 10)$ .

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Apply the distributive property.

$10 - 4 z = - - z - 1 \cdot 10$

Multiply $- - z$ .

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Multiply $- 1$ by $- 1$ .

$10 - 4 z = 1 z - 1 \cdot 10$

Multiply $z$ by $1$ .

$10 - 4 z = z - 1 \cdot 10$

Multiply $- 1$ by $10$ .

$10 - 4 z = z - 10$

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

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Subtract $z$ from both sides of the equation.

$10 - 4 z - z = - 10$

Subtract $z$ from $- 4 z$ .

$10 - 5 z = - 10$

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

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Subtract $10$ from both sides of the equation.

$- 5 z = - 10 - 10$

Subtract $10$ from $- 10$ .

$- 5 z = - 20$

Divide each term by $- 5$ and simplify.

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Divide each term in $- 5 z = - 20$ by $- 5$ .

$\frac{- 5 z}{- 5} = \frac{- 20}{- 5}$

Cancel the common factor of $- 5$ .

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Cancel the common factor.

$\frac{- 5 z}{- 5} = \frac{- 20}{- 5}$

Divide $z$ by $1$ .

$z = \frac{- 20}{- 5}$

Divide $- 20$ by $- 5$ .

$z = 4$

Consolidate the solutions.

$z = 0, 4$

Solve $2 z - 5 = - (- 2 | 10 - 4 z | + 15)$ for $z$ .

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Solve for $| 10 - 4 z |$ .

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Rewrite the equation as $- (- 2 | 10 - 4 z | + 15) = 2 z - 5$ .

$- (- 2 | 10 - 4 z | + 15) = 2 z - 5$

Multiply each term in $- (- 2 | 10 - 4 z | + 15) = 2 z - 5$ by $- 1$

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Multiply each term in $- (- 2 | 10 - 4 z | + 15) = 2 z - 5$ by $- 1$ .

$- (- 2 | 10 - 4 z | + 15) \cdot - 1 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Simplify $- (- 2 | 10 - 4 z | + 15) \cdot - 1$ .

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Apply the distributive property.

$(- (- 2 | 10 - 4 z |) - 1 \cdot 15) \cdot - 1 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Multiply.

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Multiply $- 2$ by $- 1$ .

$(2 | 10 - 4 z | - 1 \cdot 15) \cdot - 1 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Multiply $- 1$ by $15$ .

$(2 | 10 - 4 z | - 15) \cdot - 1 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Apply the distributive property.

$2 | 10 - 4 z | \cdot - 1 - 15 \cdot - 1 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Multiply.

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Multiply $- 1$ by $2$ .

$- 2 | 10 - 4 z | - 15 \cdot - 1 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Multiply $- 15$ by $- 1$ .

$- 2 | 10 - 4 z | + 15 = 2 z \cdot - 1 + (- 5) \cdot - 1$

Simplify each term.

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Multiply $- 1$ by $2$ .

$- 2 | 10 - 4 z | + 15 = - 2 z + (- 5) \cdot - 1$

Multiply $- 5$ by $- 1$ .

$- 2 | 10 - 4 z | + 15 = - 2 z + 5$

Move all terms not containing $| 10 - 4 z |$ to the right side of the equation.

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Subtract $15$ from both sides of the equation.

$- 2 | 10 - 4 z | = - 2 z + 5 - 15$

Subtract $15$ from $5$ .

$- 2 | 10 - 4 z | = - 2 z - 10$

Divide each term by $- 2$ and simplify.

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Divide each term in $- 2 | 10 - 4 z | = - 2 z - 10$ by $- 2$ .

$\frac{- 2 | 10 - 4 z |}{- 2} = \frac{- 2 z}{- 2} + \frac{- 10}{- 2}$

Cancel the common factor of $- 2$ .

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Cancel the common factor.

$\frac{- 2 | 10 - 4 z |}{- 2} = \frac{- 2 z}{- 2} + \frac{- 10}{- 2}$

Divide $| 10 - 4 z |$ by $1$ .

$| 10 - 4 z | = \frac{- 2 z}{- 2} + \frac{- 10}{- 2}$

Simplify each term.

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Cancel the common factor of $- 2$ .

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Cancel the common factor.

$| 10 - 4 z | = \frac{- 2 z}{- 2} + \frac{- 10}{- 2}$

Divide $z$ by $1$ .

$| 10 - 4 z | = z + \frac{- 10}{- 2}$

Divide $- 10$ by $- 2$ .

$| 10 - 4 z | = z + 5$

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

$10 - 4 z = \pm (z + 5)$

The result consists of both the positive and negative portions of the $\pm$ .

$10 - 4 z = z + 5$

$10 - 4 z = - (z + 5)$

Solve $10 - 4 z = z + 5$ for $z$ .

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

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Subtract $z$ from both sides of the equation.

$10 - 4 z - z = 5$

Subtract $z$ from $- 4 z$ .

$10 - 5 z = 5$

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

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Subtract $10$ from both sides of the equation.

$- 5 z = 5 - 10$

Subtract $10$ from $5$ .

$- 5 z = - 5$

Divide each term by $- 5$ and simplify.

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Divide each term in $- 5 z = - 5$ by $- 5$ .

$\frac{- 5 z}{- 5} = \frac{- 5}{- 5}$

Cancel the common factor of $- 5$ .

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Cancel the common factor.

$\frac{- 5 z}{- 5} = \frac{- 5}{- 5}$

Divide $z$ by $1$ .

$z = \frac{- 5}{- 5}$

Divide $- 5$ by $- 5$ .

$z = 1$

Solve $10 - 4 z = - (z + 5)$ for $z$ .

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Simplify $- (z + 5)$ .

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Apply the distributive property.

$10 - 4 z = - z - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$10 - 4 z = - z - 5$

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

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Add $z$ to both sides of the equation.

$10 - 4 z + z = - 5$

Add $- 4 z$ and $z$ .

$10 - 3 z = - 5$

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

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Subtract $10$ from both sides of the equation.

$- 3 z = - 5 - 10$

Subtract $10$ from $- 5$ .

$- 3 z = - 15$

Divide each term by $- 3$ and simplify.

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Divide each term in $- 3 z = - 15$ by $- 3$ .

$\frac{- 3 z}{- 3} = \frac{- 15}{- 3}$

Cancel the common factor of $- 3$ .

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Cancel the common factor.

$\frac{- 3 z}{- 3} = \frac{- 15}{- 3}$

Divide $z$ by $1$ .

$z = \frac{- 15}{- 3}$

Divide $- 15$ by $- 3$ .

$z = 5$

Consolidate the solutions.

$z = 1, 5$

Consolidate the solutions.

$z = 0, 4, 1, 5$

Exclude the solutions that do not make $| 2 z - 5 | + 2 | 10 - 4 z | = 15$ true.

$z = 4, 1$

Simplify |2z-5|+2|10-4z|=15