Basic Math Examples

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

Step 1

Subtract $| 2 z - 5 |$ from both sides of the equation.

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

Step 2

Divide each term in $2 | 10 - 4 z | = 15 - | 2 z - 5 |$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 | 10 - 4 z | = 15 - | 2 z - 5 |$ by $2$ .

$\frac{2 | 10 - 4 z |}{2} = \frac{15}{2} + \frac{- | 2 z - 5 |}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 | 10 - 4 z |}{2} = \frac{15}{2} + \frac{- | 2 z - 5 |}{2}$

Step 2.2.1.2

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

$| 10 - 4 z | = \frac{15}{2} + \frac{- | 2 z - 5 |}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$| 10 - 4 z | = \frac{15}{2} - \frac{| 2 z - 5 |}{2}$

Step 3

Solve for $| 2 z - 5 |$ .

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Step 3.1

Rewrite the equation as $\frac{15}{2} - \frac{| 2 z - 5 |}{2} = | 10 - 4 z |$ .

$\frac{15}{2} - \frac{| 2 z - 5 |}{2} = | 10 - 4 z |$

Step 3.2

Subtract $\frac{15}{2}$ from both sides of the equation.

$- \frac{| 2 z - 5 |}{2} = | 10 - 4 z | - \frac{15}{2}$

Step 3.3

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{| 2 z - 5 |}{2}) = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $- 2 (- \frac{| 2 z - 5 |}{2})$ .

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Step 3.4.1.1.1

Cancel the common factor of $2$ .

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Step 3.4.1.1.1.1

Move the leading negative in $- \frac{| 2 z - 5 |}{2}$ into the numerator.

$- 2 \frac{- | 2 z - 5 |}{2} = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- | 2 z - 5 |}{2} = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- | 2 z - 5 |}{2} = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4.1.1.1.4

Rewrite the expression.

$- - | 2 z - 5 | = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4.1.1.2

Multiply.

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Step 3.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 | 2 z - 5 | = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4.1.1.2.2

Multiply $| 2 z - 5 |$ by $1$ .

$| 2 z - 5 | = - 2 (| 10 - 4 z | - \frac{15}{2})$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 2 (| 10 - 4 z | - \frac{15}{2})$ .

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Step 3.4.2.1.1

Apply the distributive property.

$| 2 z - 5 | = - 2 | 10 - 4 z | - 2 (- \frac{15}{2})$

Step 3.4.2.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.2.1

Move the leading negative in $- \frac{15}{2}$ into the numerator.

$| 2 z - 5 | = - 2 | 10 - 4 z | - 2 (\frac{- 15}{2})$

Step 3.4.2.1.2.2

Factor $2$ out of $- 2$ .

$| 2 z - 5 | = - 2 | 10 - 4 z | + 2 (- 1) \frac{- 15}{2}$

Step 3.4.2.1.2.3

Cancel the common factor.

$| 2 z - 5 | = - 2 | 10 - 4 z | + 2 \cdot - 1 \frac{- 15}{2}$

Step 3.4.2.1.2.4

Rewrite the expression.

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

Step 3.4.2.1.3

Multiply $- 1$ by $- 15$ .

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

Step 4

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

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

Step 5

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

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

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

Step 6

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

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Step 6.1

Solve for $| 10 - 4 z |$ .

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Step 6.1.1

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

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

Step 6.1.2

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

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Step 6.1.2.1

Subtract $15$ from both sides of the equation.

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

Step 6.1.2.2

Subtract $15$ from $- 5$ .

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

Step 6.1.3

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

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Step 6.1.3.1

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

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

Step 6.1.3.2

Simplify the left side.

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Step 6.1.3.2.1

Cancel the common factor of $- 2$ .

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Step 6.1.3.2.1.1

Cancel the common factor.

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

Step 6.1.3.2.1.2

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

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

Step 6.1.3.3

Simplify the right side.

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Step 6.1.3.3.1

Simplify each term.

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Step 6.1.3.3.1.1

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

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Step 6.1.3.3.1.1.1

Factor $2$ out of $2 z$ .

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

Step 6.1.3.3.1.1.2

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

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

Step 6.1.3.3.1.2

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

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

Step 6.1.3.3.1.3

Divide $- 20$ by $- 2$ .

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

Step 6.2

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

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

Step 6.3

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

$10 - 4 z = - z + 10$

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

Step 6.4

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

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Step 6.4.1

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

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Step 6.4.1.1

Add $z$ to both sides of the equation.

$10 - 4 z + z = 10$

Step 6.4.1.2

Add $- 4 z$ and $z$ .

$10 - 3 z = 10$

Step 6.4.2

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

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Step 6.4.2.1

Subtract $10$ from both sides of the equation.

$- 3 z = 10 - 10$

Step 6.4.2.2

Subtract $10$ from $10$ .

$- 3 z = 0$

Step 6.4.3

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

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Step 6.4.3.1

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

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

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Cancel the common factor of $- 3$ .

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Step 6.4.3.2.1.1

Cancel the common factor.

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

Step 6.4.3.2.1.2

Divide $z$ by $1$ .

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

Step 6.4.3.3

Simplify the right side.

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Step 6.4.3.3.1

Divide $0$ by $- 3$ .

$z = 0$

Step 6.5

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

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Step 6.5.1

Simplify $- (- z + 10)$ .

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Step 6.5.1.1

Rewrite.

$10 - 4 z = 0 + 0 - (- z + 10)$

Step 6.5.1.2

Simplify by adding zeros.

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

Step 6.5.1.3

Apply the distributive property.

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

Step 6.5.1.4

Multiply $- - z$ .

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Step 6.5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.5.1.4.2

Multiply $z$ by $1$ .

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

Step 6.5.1.5

Multiply $- 1$ by $10$ .

$10 - 4 z = z - 10$

Step 6.5.2

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

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Step 6.5.2.1

Subtract $z$ from both sides of the equation.

$10 - 4 z - z = - 10$

Step 6.5.2.2

Subtract $z$ from $- 4 z$ .

$10 - 5 z = - 10$

Step 6.5.3

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

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Step 6.5.3.1

Subtract $10$ from both sides of the equation.

$- 5 z = - 10 - 10$

Step 6.5.3.2

Subtract $10$ from $- 10$ .

$- 5 z = - 20$

Step 6.5.4

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

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Step 6.5.4.1

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

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

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $- 5$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

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

Step 6.5.4.2.1.2

Divide $z$ by $1$ .

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

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Divide $- 20$ by $- 5$ .

$z = 4$

Step 6.6

Consolidate the solutions.

$z = 0, 4$

Step 7

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

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Step 7.1

Solve for $| 10 - 4 z |$ .

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

Apply the distributive property.

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

Step 7.1.2.2

Multiply.

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Step 7.1.2.2.1

Multiply $- 2$ by $- 1$ .

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

Step 7.1.2.2.2

Multiply $- 1$ by $15$ .

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

Step 7.1.3

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

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Step 7.1.3.1

Add $15$ to both sides of the equation.

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

Step 7.1.3.2

Add $- 5$ and $15$ .

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

Step 7.1.4

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

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Step 7.1.4.1

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

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

Step 7.1.4.2

Simplify the left side.

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Step 7.1.4.2.1

Cancel the common factor of $2$ .

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Step 7.1.4.2.1.1

Cancel the common factor.

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

Step 7.1.4.2.1.2

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

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

Step 7.1.4.3

Simplify the right side.

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Step 7.1.4.3.1

Simplify each term.

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Step 7.1.4.3.1.1

Cancel the common factor of $2$ .

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Step 7.1.4.3.1.1.1

Cancel the common factor.

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

Step 7.1.4.3.1.1.2

Divide $z$ by $1$ .

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

Step 7.1.4.3.1.2

Divide $10$ by $2$ .

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

Step 7.2

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

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

Step 7.3

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

$10 - 4 z = z + 5$

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

Step 7.4

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

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Step 7.4.1

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

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Step 7.4.1.1

Subtract $z$ from both sides of the equation.

$10 - 4 z - z = 5$

Step 7.4.1.2

Subtract $z$ from $- 4 z$ .

$10 - 5 z = 5$

Step 7.4.2

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

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Step 7.4.2.1

Subtract $10$ from both sides of the equation.

$- 5 z = 5 - 10$

Step 7.4.2.2

Subtract $10$ from $5$ .

$- 5 z = - 5$

Step 7.4.3

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

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Step 7.4.3.1

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

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

Step 7.4.3.2

Simplify the left side.

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Step 7.4.3.2.1

Cancel the common factor of $- 5$ .

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Step 7.4.3.2.1.1

Cancel the common factor.

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

Step 7.4.3.2.1.2

Divide $z$ by $1$ .

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

Step 7.4.3.3

Simplify the right side.

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Step 7.4.3.3.1

Divide $- 5$ by $- 5$ .

$z = 1$

Step 7.5

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

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Step 7.5.1

Simplify $- (z + 5)$ .

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Step 7.5.1.1

Rewrite.

$10 - 4 z = 0 + 0 - (z + 5)$

Step 7.5.1.2

Simplify by adding zeros.

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

Step 7.5.1.3

Apply the distributive property.

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

Step 7.5.1.4

Multiply $- 1$ by $5$ .

$10 - 4 z = - z - 5$

Step 7.5.2

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

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Step 7.5.2.1

Add $z$ to both sides of the equation.

$10 - 4 z + z = - 5$

Step 7.5.2.2

Add $- 4 z$ and $z$ .

$10 - 3 z = - 5$

Step 7.5.3

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

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Step 7.5.3.1

Subtract $10$ from both sides of the equation.

$- 3 z = - 5 - 10$

Step 7.5.3.2

Subtract $10$ from $- 5$ .

$- 3 z = - 15$

Step 7.5.4

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

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Step 7.5.4.1

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

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

Step 7.5.4.2

Simplify the left side.

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Step 7.5.4.2.1

Cancel the common factor of $- 3$ .

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Step 7.5.4.2.1.1

Cancel the common factor.

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

Step 7.5.4.2.1.2

Divide $z$ by $1$ .

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

Step 7.5.4.3

Simplify the right side.

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Step 7.5.4.3.1

Divide $- 15$ by $- 3$ .

$z = 5$

Step 7.6

Consolidate the solutions.

$z = 1, 5$

Step 8

Consolidate the solutions.

$z = 0, 4, 1, 5$

Step 9

Use each root to create test intervals.

$z < 0$

$0 < z < 1$

$1 < z < 4$

$4 < z < 5$

$z > 5$

Step 10

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 10.1

Test a value on the interval $z < 0$ to see if it makes the inequality true.

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Step 10.1.1

Choose a value on the interval $z < 0$ and see if this value makes the original inequality true.

$z = - 2$

Step 10.1.2

Replace $z$ with $- 2$ in the original inequality.

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

Step 10.1.3

The left side $45$ does not equal to the right side $15$ , which means that the given statement is false.

False

Step 10.2

Test a value on the interval $0 < z < 1$ to see if it makes the inequality true.

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Step 10.2.1

Choose a value on the interval $0 < z < 1$ and see if this value makes the original inequality true.

$z = 0.5$

Step 10.2.2

Replace $z$ with $0.5$ in the original inequality.

$| 2 (0.5) - 5 | + 2 | 10 - 4 \cdot 0.5 | = 15$

Step 10.2.3

The left side $20$ does not equal to the right side $15$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $1 < z < 4$ to see if it makes the inequality true.

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Step 10.3.1

Choose a value on the interval $1 < z < 4$ and see if this value makes the original inequality true.

$z = 2$

Step 10.3.2

Replace $z$ with $2$ in the original inequality.

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

Step 10.3.3

The left side $5$ does not equal to the right side $15$ , which means that the given statement is false.

False

Step 10.4

Test a value on the interval $4 < z < 5$ to see if it makes the inequality true.

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Step 10.4.1

Choose a value on the interval $4 < z < 5$ and see if this value makes the original inequality true.

$z = 4.5$

Step 10.4.2

Replace $z$ with $4.5$ in the original inequality.

$| 2 (4.5) - 5 | + 2 | 10 - 4 \cdot 4.5 | = 15$

Step 10.4.3

The left side $20$ does not equal to the right side $15$ , which means that the given statement is false.

False

Step 10.5

Test a value on the interval $z > 5$ to see if it makes the inequality true.

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Step 10.5.1

Choose a value on the interval $z > 5$ and see if this value makes the original inequality true.

$z = 8$

Step 10.5.2

Replace $z$ with $8$ in the original inequality.

$| 2 (8) - 5 | + 2 | 10 - 4 \cdot 8 | = 15$

Step 10.5.3

The left side $55$ does not equal to the right side $15$ , which means that the given statement is false.

False

Step 10.6

Compare the intervals to determine which ones satisfy the original inequality.

$z < 0$ False

$0 < z < 1$ False

$1 < z < 4$ False

$4 < z < 5$ False

$z > 5$ False

$z < 0$ False

$0 < z < 1$ False

$1 < z < 4$ False

$4 < z < 5$ False

$z > 5$ False

Step 11

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 12

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

$z = 4, 1$

Step 13