Finite Math Examples

$| x - 2 | + | 1 - 5 x | + | 2 x - 1 | = 3 x$

Step 1

Move all terms not containing $| 2 x - 1 |$ to the right side of the equation.

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Step 1.1

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

$| 1 - 5 x | + | 2 x - 1 | = 3 x - | x - 2 |$

Step 1.2

Subtract $| 1 - 5 x |$ from both sides of the equation.

$| 2 x - 1 | = 3 x - | x - 2 | - | 1 - 5 x |$

Step 2

Solve for $| 1 - 5 x |$ .

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Step 2.1

Rewrite the equation as $3 x - | x - 2 | - | 1 - 5 x | = | 2 x - 1 |$ .

$3 x - | x - 2 | - | 1 - 5 x | = | 2 x - 1 |$

Step 2.2

Move all terms not containing $| 1 - 5 x |$ to the right side of the equation.

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Step 2.2.1

Subtract $3 x$ from both sides of the equation.

$- | x - 2 | - | 1 - 5 x | = | 2 x - 1 | - 3 x$

Step 2.2.2

Add $| x - 2 |$ to both sides of the equation.

$- | 1 - 5 x | = | 2 x - 1 | - 3 x + | x - 2 |$

Step 2.3

Divide each term in $- | 1 - 5 x | = | 2 x - 1 | - 3 x + | x - 2 |$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- | 1 - 5 x | = | 2 x - 1 | - 3 x + | x - 2 |$ by $- 1$ .

$\frac{- | 1 - 5 x |}{- 1} = \frac{| 2 x - 1 |}{- 1} + \frac{- 3 x}{- 1} + \frac{| x - 2 |}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{| 1 - 5 x |}{1} = \frac{| 2 x - 1 |}{- 1} + \frac{- 3 x}{- 1} + \frac{| x - 2 |}{- 1}$

Step 2.3.2.2

Divide $| 1 - 5 x |$ by $1$ .

$| 1 - 5 x | = \frac{| 2 x - 1 |}{- 1} + \frac{- 3 x}{- 1} + \frac{| x - 2 |}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{| 2 x - 1 |}{- 1}$ .

$| 1 - 5 x | = - 1 \cdot | 2 x - 1 | + \frac{- 3 x}{- 1} + \frac{| x - 2 |}{- 1}$

Step 2.3.3.1.2

Rewrite $- 1 \cdot | 2 x - 1 |$ as $- | 2 x - 1 |$ .

$| 1 - 5 x | = - | 2 x - 1 | + \frac{- 3 x}{- 1} + \frac{| x - 2 |}{- 1}$

Step 2.3.3.1.3

Move the negative one from the denominator of $\frac{- 3 x}{- 1}$ .

$| 1 - 5 x | = - | 2 x - 1 | - 1 \cdot (- 3 x) + \frac{| x - 2 |}{- 1}$

Step 2.3.3.1.4

Rewrite $- 1 \cdot (- 3 x)$ as $- (- 3 x)$ .

$| 1 - 5 x | = - | 2 x - 1 | - (- 3 x) + \frac{| x - 2 |}{- 1}$

Step 2.3.3.1.5

Multiply $- 3$ by $- 1$ .

$| 1 - 5 x | = - | 2 x - 1 | + 3 x + \frac{| x - 2 |}{- 1}$

Step 2.3.3.1.6

Move the negative one from the denominator of $\frac{| x - 2 |}{- 1}$ .

$| 1 - 5 x | = - | 2 x - 1 | + 3 x - 1 \cdot | x - 2 |$

Step 2.3.3.1.7

Rewrite $- 1 \cdot | x - 2 |$ as $- | x - 2 |$ .

$| 1 - 5 x | = - | 2 x - 1 | + 3 x - | x - 2 |$

Step 3

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

$1 - 5 x = \pm (- | 2 x - 1 | + 3 x - | x - 2 |)$

Step 4

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

$1 - 5 x = - | 2 x - 1 | + 3 x - | x - 2 |$

$1 - 5 x = - (- | 2 x - 1 | + 3 x - | x - 2 |)$

Step 5

Solve $1 - 5 x = - | 2 x - 1 | + 3 x - | x - 2 |$ for $x$ .

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Step 5.1

Solve for $| x - 2 |$ .

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Step 5.1.1

Rewrite the equation as $- | 2 x - 1 | + 3 x - | x - 2 | = 1 - 5 x$ .

$- | 2 x - 1 | + 3 x - | x - 2 | = 1 - 5 x$

Step 5.1.2

Move all terms not containing $| x - 2 |$ to the right side of the equation.

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Step 5.1.2.1

Add $| 2 x - 1 |$ to both sides of the equation.

$3 x - | x - 2 | = 1 - 5 x + | 2 x - 1 |$

Step 5.1.2.2

Subtract $3 x$ from both sides of the equation.

$- | x - 2 | = 1 - 5 x + | 2 x - 1 | - 3 x$

Step 5.1.2.3

Subtract $3 x$ from $- 5 x$ .

$- | x - 2 | = 1 + | 2 x - 1 | - 8 x$

Step 5.1.3

Divide each term in $- | x - 2 | = 1 + | 2 x - 1 | - 8 x$ by $- 1$ and simplify.

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Step 5.1.3.1

Divide each term in $- | x - 2 | = 1 + | 2 x - 1 | - 8 x$ by $- 1$ .

$\frac{- | x - 2 |}{- 1} = \frac{1}{- 1} + \frac{| 2 x - 1 |}{- 1} + \frac{- 8 x}{- 1}$

Step 5.1.3.2

Simplify the left side.

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Step 5.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 2 |}{1} = \frac{1}{- 1} + \frac{| 2 x - 1 |}{- 1} + \frac{- 8 x}{- 1}$

Step 5.1.3.2.2

Divide $| x - 2 |$ by $1$ .

$| x - 2 | = \frac{1}{- 1} + \frac{| 2 x - 1 |}{- 1} + \frac{- 8 x}{- 1}$

Step 5.1.3.3

Simplify the right side.

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Step 5.1.3.3.1

Simplify each term.

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Step 5.1.3.3.1.1

Divide $1$ by $- 1$ .

$| x - 2 | = - 1 + \frac{| 2 x - 1 |}{- 1} + \frac{- 8 x}{- 1}$

Step 5.1.3.3.1.2

Move the negative one from the denominator of $\frac{| 2 x - 1 |}{- 1}$ .

$| x - 2 | = - 1 - 1 \cdot | 2 x - 1 | + \frac{- 8 x}{- 1}$

Step 5.1.3.3.1.3

Rewrite $- 1 \cdot | 2 x - 1 |$ as $- | 2 x - 1 |$ .

$| x - 2 | = - 1 - | 2 x - 1 | + \frac{- 8 x}{- 1}$

Step 5.1.3.3.1.4

Move the negative one from the denominator of $\frac{- 8 x}{- 1}$ .

$| x - 2 | = - 1 - | 2 x - 1 | - 1 \cdot (- 8 x)$

Step 5.1.3.3.1.5

Rewrite $- 1 \cdot (- 8 x)$ as $- (- 8 x)$ .

$| x - 2 | = - 1 - | 2 x - 1 | - (- 8 x)$

Step 5.1.3.3.1.6

Multiply $- 8$ by $- 1$ .

$| x - 2 | = - 1 - | 2 x - 1 | + 8 x$

Step 5.2

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

$x - 2 = \pm (- 1 - | 2 x - 1 | + 8 x)$

Step 5.3

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

$x - 2 = - 1 - | 2 x - 1 | + 8 x$

$x - 2 = - (- 1 - | 2 x - 1 | + 8 x)$

Step 5.4

Solve $x - 2 = - 1 - | 2 x - 1 | + 8 x$ for $x$ .

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Step 5.4.1

Solve for $| 2 x - 1 |$ .

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Step 5.4.1.1

Rewrite the equation as $- 1 - | 2 x - 1 | + 8 x = x - 2$ .

$- 1 - | 2 x - 1 | + 8 x = x - 2$

Step 5.4.1.2

Move all terms not containing $| 2 x - 1 |$ to the right side of the equation.

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Step 5.4.1.2.1

Add $1$ to both sides of the equation.

$- | 2 x - 1 | + 8 x = x - 2 + 1$

Step 5.4.1.2.2

Subtract $8 x$ from both sides of the equation.

$- | 2 x - 1 | = x - 2 + 1 - 8 x$

Step 5.4.1.2.3

Subtract $8 x$ from $x$ .

$- | 2 x - 1 | = - 7 x - 2 + 1$

Step 5.4.1.2.4

Add $- 2$ and $1$ .

$- | 2 x - 1 | = - 7 x - 1$

Step 5.4.1.3

Divide each term in $- | 2 x - 1 | = - 7 x - 1$ by $- 1$ and simplify.

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Step 5.4.1.3.1

Divide each term in $- | 2 x - 1 | = - 7 x - 1$ by $- 1$ .

$\frac{- | 2 x - 1 |}{- 1} = \frac{- 7 x}{- 1} + \frac{- 1}{- 1}$

Step 5.4.1.3.2

Simplify the left side.

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Step 5.4.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 x - 1 |}{1} = \frac{- 7 x}{- 1} + \frac{- 1}{- 1}$

Step 5.4.1.3.2.2

Divide $| 2 x - 1 |$ by $1$ .

$| 2 x - 1 | = \frac{- 7 x}{- 1} + \frac{- 1}{- 1}$

Step 5.4.1.3.3

Simplify the right side.

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Step 5.4.1.3.3.1

Simplify each term.

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Step 5.4.1.3.3.1.1

Move the negative one from the denominator of $\frac{- 7 x}{- 1}$ .

$| 2 x - 1 | = - 1 \cdot (- 7 x) + \frac{- 1}{- 1}$

Step 5.4.1.3.3.1.2

Rewrite $- 1 \cdot (- 7 x)$ as $- (- 7 x)$ .

$| 2 x - 1 | = - (- 7 x) + \frac{- 1}{- 1}$

Step 5.4.1.3.3.1.3

Multiply $- 7$ by $- 1$ .

$| 2 x - 1 | = 7 x + \frac{- 1}{- 1}$

Step 5.4.1.3.3.1.4

Divide $- 1$ by $- 1$ .

$| 2 x - 1 | = 7 x + 1$

Step 5.4.2

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

$2 x - 1 = \pm (7 x + 1)$

Step 5.4.3

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

$2 x - 1 = 7 x + 1$

$2 x - 1 = - (7 x + 1)$

Step 5.4.4

Solve $2 x - 1 = 7 x + 1$ for $x$ .

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Step 5.4.4.1

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

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Step 5.4.4.1.1

Subtract $7 x$ from both sides of the equation.

$2 x - 1 - 7 x = 1$

Step 5.4.4.1.2

Subtract $7 x$ from $2 x$ .

$- 5 x - 1 = 1$

Step 5.4.4.2

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

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Step 5.4.4.2.1

Add $1$ to both sides of the equation.

$- 5 x = 1 + 1$

Step 5.4.4.2.2

Add $1$ and $1$ .

$- 5 x = 2$

Step 5.4.4.3

Divide each term in $- 5 x = 2$ by $- 5$ and simplify.

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Step 5.4.4.3.1

Divide each term in $- 5 x = 2$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{2}{- 5}$

Step 5.4.4.3.2

Simplify the left side.

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Step 5.4.4.3.2.1

Cancel the common factor of $- 5$ .

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Step 5.4.4.3.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{2}{- 5}$

Step 5.4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{- 5}$

Step 5.4.4.3.3

Simplify the right side.

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Step 5.4.4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 5.4.5

Solve $2 x - 1 = - (7 x + 1)$ for $x$ .

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Step 5.4.5.1

Simplify $- (7 x + 1)$ .

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Step 5.4.5.1.1

Rewrite.

$2 x - 1 = 0 + 0 - (7 x + 1)$

Step 5.4.5.1.2

Simplify by adding zeros.

$2 x - 1 = - (7 x + 1)$

Step 5.4.5.1.3

Apply the distributive property.

$2 x - 1 = - (7 x) - 1 \cdot 1$

Step 5.4.5.1.4

Multiply.

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Step 5.4.5.1.4.1

Multiply $7$ by $- 1$ .

$2 x - 1 = - 7 x - 1 \cdot 1$

Step 5.4.5.1.4.2

Multiply $- 1$ by $1$ .

$2 x - 1 = - 7 x - 1$

Step 5.4.5.2

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

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Step 5.4.5.2.1

Add $7 x$ to both sides of the equation.

$2 x - 1 + 7 x = - 1$

Step 5.4.5.2.2

Add $2 x$ and $7 x$ .

$9 x - 1 = - 1$

Step 5.4.5.3

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

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Step 5.4.5.3.1

Add $1$ to both sides of the equation.

$9 x = - 1 + 1$

Step 5.4.5.3.2

Add $- 1$ and $1$ .

$9 x = 0$

Step 5.4.5.4

Divide each term in $9 x = 0$ by $9$ and simplify.

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Step 5.4.5.4.1

Divide each term in $9 x = 0$ by $9$ .

$\frac{9 x}{9} = \frac{0}{9}$

Step 5.4.5.4.2

Simplify the left side.

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Step 5.4.5.4.2.1

Cancel the common factor of $9$ .

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Step 5.4.5.4.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{0}{9}$

Step 5.4.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{9}$

Step 5.4.5.4.3

Simplify the right side.

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Step 5.4.5.4.3.1

Divide $0$ by $9$ .

$x = 0$

Step 5.4.6

Consolidate the solutions.

$x = - \frac{2}{5}, 0$

Step 5.5

Solve $x - 2 = - (- 1 - | 2 x - 1 | + 8 x)$ for $x$ .

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Step 5.5.1

Solve for $| 2 x - 1 |$ .

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Step 5.5.1.1

Rewrite the equation as $- (- 1 - | 2 x - 1 | + 8 x) = x - 2$ .

$- (- 1 - | 2 x - 1 | + 8 x) = x - 2$

Step 5.5.1.2

Simplify $- (- 1 - | 2 x - 1 | + 8 x)$ .

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Step 5.5.1.2.1

Apply the distributive property.

$- - 1 - - | 2 x - 1 | - (8 x) = x - 2$

Step 5.5.1.2.2

Simplify.

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Step 5.5.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 - - | 2 x - 1 | - (8 x) = x - 2$

Step 5.5.1.2.2.2

Multiply $- - | 2 x - 1 |$ .

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Step 5.5.1.2.2.2.1

Multiply $- 1$ by $- 1$ .

$1 + 1 | 2 x - 1 | - (8 x) = x - 2$

Step 5.5.1.2.2.2.2

Multiply $| 2 x - 1 |$ by $1$ .

$1 + | 2 x - 1 | - (8 x) = x - 2$

Step 5.5.1.2.2.3

Multiply $8$ by $- 1$ .

$1 + | 2 x - 1 | - 8 x = x - 2$

Step 5.5.1.3

Move all terms not containing $| 2 x - 1 |$ to the right side of the equation.

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Step 5.5.1.3.1

Subtract $1$ from both sides of the equation.

$| 2 x - 1 | - 8 x = x - 2 - 1$

Step 5.5.1.3.2

Add $8 x$ to both sides of the equation.

$| 2 x - 1 | = x - 2 - 1 + 8 x$

Step 5.5.1.3.3

Add $x$ and $8 x$ .

$| 2 x - 1 | = 9 x - 2 - 1$

Step 5.5.1.3.4

Subtract $1$ from $- 2$ .

$| 2 x - 1 | = 9 x - 3$

Step 5.5.2

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

$2 x - 1 = \pm (9 x - 3)$

Step 5.5.3

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

$2 x - 1 = 9 x - 3$

$2 x - 1 = - (9 x - 3)$

Step 5.5.4

Solve $2 x - 1 = 9 x - 3$ for $x$ .

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Step 5.5.4.1

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

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Step 5.5.4.1.1

Subtract $9 x$ from both sides of the equation.

$2 x - 1 - 9 x = - 3$

Step 5.5.4.1.2

Subtract $9 x$ from $2 x$ .

$- 7 x - 1 = - 3$

Step 5.5.4.2

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

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Step 5.5.4.2.1

Add $1$ to both sides of the equation.

$- 7 x = - 3 + 1$

Step 5.5.4.2.2

Add $- 3$ and $1$ .

$- 7 x = - 2$

Step 5.5.4.3

Divide each term in $- 7 x = - 2$ by $- 7$ and simplify.

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Step 5.5.4.3.1

Divide each term in $- 7 x = - 2$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 2}{- 7}$

Step 5.5.4.3.2

Simplify the left side.

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Step 5.5.4.3.2.1

Cancel the common factor of $- 7$ .

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Step 5.5.4.3.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 2}{- 7}$

Step 5.5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 7}$

Step 5.5.4.3.3

Simplify the right side.

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Step 5.5.4.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{2}{7}$

Step 5.5.5

Solve $2 x - 1 = - (9 x - 3)$ for $x$ .

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Step 5.5.5.1

Simplify $- (9 x - 3)$ .

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Step 5.5.5.1.1

Rewrite.

$2 x - 1 = 0 + 0 - (9 x - 3)$

Step 5.5.5.1.2

Simplify by adding zeros.

$2 x - 1 = - (9 x - 3)$

Step 5.5.5.1.3

Apply the distributive property.

$2 x - 1 = - (9 x) - - 3$

Step 5.5.5.1.4

Multiply.

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Step 5.5.5.1.4.1

Multiply $9$ by $- 1$ .

$2 x - 1 = - 9 x - - 3$

Step 5.5.5.1.4.2

Multiply $- 1$ by $- 3$ .

$2 x - 1 = - 9 x + 3$

Step 5.5.5.2

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

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Step 5.5.5.2.1

Add $9 x$ to both sides of the equation.

$2 x - 1 + 9 x = 3$

Step 5.5.5.2.2

Add $2 x$ and $9 x$ .

$11 x - 1 = 3$

Step 5.5.5.3

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

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Step 5.5.5.3.1

Add $1$ to both sides of the equation.

$11 x = 3 + 1$

Step 5.5.5.3.2

Add $3$ and $1$ .

$11 x = 4$

Step 5.5.5.4

Divide each term in $11 x = 4$ by $11$ and simplify.

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Step 5.5.5.4.1

Divide each term in $11 x = 4$ by $11$ .

$\frac{11 x}{11} = \frac{4}{11}$

Step 5.5.5.4.2

Simplify the left side.

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Step 5.5.5.4.2.1

Cancel the common factor of $11$ .

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Step 5.5.5.4.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{4}{11}$

Step 5.5.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{11}$

Step 5.5.6

Consolidate the solutions.

$x = \frac{2}{7}, \frac{4}{11}$

Step 5.6

Consolidate the solutions.

$x = - \frac{2}{5}, 0, \frac{2}{7}, \frac{4}{11}$

Step 6

Solve $1 - 5 x = - (- | 2 x - 1 | + 3 x - | x - 2 |)$ for $x$ .

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Step 6.1

Solve for $| x - 2 |$ .

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Step 6.1.1

Rewrite the equation as $- (- | 2 x - 1 | + 3 x - | x - 2 |) = 1 - 5 x$ .

$- (- | 2 x - 1 | + 3 x - | x - 2 |) = 1 - 5 x$

Step 6.1.2

Simplify $- (- | 2 x - 1 | + 3 x - | x - 2 |)$ .

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Step 6.1.2.1

Apply the distributive property.

$- - | 2 x - 1 | - (3 x) - - | x - 2 | = 1 - 5 x$

Step 6.1.2.2

Simplify.

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Step 6.1.2.2.1

Multiply $- - | 2 x - 1 |$ .

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Step 6.1.2.2.1.1

Multiply $- 1$ by $- 1$ .

$1 | 2 x - 1 | - (3 x) - - | x - 2 | = 1 - 5 x$

Step 6.1.2.2.1.2

Multiply $| 2 x - 1 |$ by $1$ .

$| 2 x - 1 | - (3 x) - - | x - 2 | = 1 - 5 x$

Step 6.1.2.2.2

Multiply $3$ by $- 1$ .

$| 2 x - 1 | - 3 x - - | x - 2 | = 1 - 5 x$

Step 6.1.2.2.3

Multiply $- - | x - 2 |$ .

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Step 6.1.2.2.3.1

Multiply $- 1$ by $- 1$ .

$| 2 x - 1 | - 3 x + 1 | x - 2 | = 1 - 5 x$

Step 6.1.2.2.3.2

Multiply $| x - 2 |$ by $1$ .

$| 2 x - 1 | - 3 x + | x - 2 | = 1 - 5 x$

Step 6.1.3

Move all terms not containing $| x - 2 |$ to the right side of the equation.

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Step 6.1.3.1

Subtract $| 2 x - 1 |$ from both sides of the equation.

$- 3 x + | x - 2 | = 1 - 5 x - | 2 x - 1 |$

Step 6.1.3.2

Add $3 x$ to both sides of the equation.

$| x - 2 | = 1 - 5 x - | 2 x - 1 | + 3 x$

Step 6.1.3.3

Add $- 5 x$ and $3 x$ .

$| x - 2 | = 1 - | 2 x - 1 | - 2 x$

Step 6.2

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

$x - 2 = \pm (1 - | 2 x - 1 | - 2 x)$

Step 6.3

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

$x - 2 = 1 - | 2 x - 1 | - 2 x$

$x - 2 = - (1 - | 2 x - 1 | - 2 x)$

Step 6.4

Solve $x - 2 = 1 - | 2 x - 1 | - 2 x$ for $x$ .

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Step 6.4.1

Solve for $| 2 x - 1 |$ .

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Step 6.4.1.1

Rewrite the equation as $1 - | 2 x - 1 | - 2 x = x - 2$ .

$1 - | 2 x - 1 | - 2 x = x - 2$

Step 6.4.1.2

Move all terms not containing $| 2 x - 1 |$ to the right side of the equation.

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Step 6.4.1.2.1

Subtract $1$ from both sides of the equation.

$- | 2 x - 1 | - 2 x = x - 2 - 1$

Step 6.4.1.2.2

Add $2 x$ to both sides of the equation.

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

Step 6.4.1.2.3

Add $x$ and $2 x$ .

$- | 2 x - 1 | = 3 x - 2 - 1$

Step 6.4.1.2.4

Subtract $1$ from $- 2$ .

$- | 2 x - 1 | = 3 x - 3$

Step 6.4.1.3

Divide each term in $- | 2 x - 1 | = 3 x - 3$ by $- 1$ and simplify.

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Step 6.4.1.3.1

Divide each term in $- | 2 x - 1 | = 3 x - 3$ by $- 1$ .

$\frac{- | 2 x - 1 |}{- 1} = \frac{3 x}{- 1} + \frac{- 3}{- 1}$

Step 6.4.1.3.2

Simplify the left side.

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Step 6.4.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 x - 1 |}{1} = \frac{3 x}{- 1} + \frac{- 3}{- 1}$

Step 6.4.1.3.2.2

Divide $| 2 x - 1 |$ by $1$ .

$| 2 x - 1 | = \frac{3 x}{- 1} + \frac{- 3}{- 1}$

Step 6.4.1.3.3

Simplify the right side.

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Step 6.4.1.3.3.1

Simplify each term.

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Step 6.4.1.3.3.1.1

Move the negative one from the denominator of $\frac{3 x}{- 1}$ .

$| 2 x - 1 | = - 1 \cdot (3 x) + \frac{- 3}{- 1}$

Step 6.4.1.3.3.1.2

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

$| 2 x - 1 | = - (3 x) + \frac{- 3}{- 1}$

Step 6.4.1.3.3.1.3

Multiply $3$ by $- 1$ .

$| 2 x - 1 | = - 3 x + \frac{- 3}{- 1}$

Step 6.4.1.3.3.1.4

Divide $- 3$ by $- 1$ .

$| 2 x - 1 | = - 3 x + 3$

Step 6.4.2

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

$2 x - 1 = \pm (- 3 x + 3)$

Step 6.4.3

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

$2 x - 1 = - 3 x + 3$

$2 x - 1 = - (- 3 x + 3)$

Step 6.4.4

Solve $2 x - 1 = - 3 x + 3$ for $x$ .

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Step 6.4.4.1

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

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Step 6.4.4.1.1

Add $3 x$ to both sides of the equation.

$2 x - 1 + 3 x = 3$

Step 6.4.4.1.2

Add $2 x$ and $3 x$ .

$5 x - 1 = 3$

Step 6.4.4.2

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

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Step 6.4.4.2.1

Add $1$ to both sides of the equation.

$5 x = 3 + 1$

Step 6.4.4.2.2

Add $3$ and $1$ .

$5 x = 4$

Step 6.4.4.3

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 6.4.4.3.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 6.4.4.3.2

Simplify the left side.

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Step 6.4.4.3.2.1

Cancel the common factor of $5$ .

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Step 6.4.4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 6.4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 6.4.5

Solve $2 x - 1 = - (- 3 x + 3)$ for $x$ .

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Step 6.4.5.1

Simplify $- (- 3 x + 3)$ .

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Step 6.4.5.1.1

Rewrite.

$2 x - 1 = 0 + 0 - (- 3 x + 3)$

Step 6.4.5.1.2

Simplify by adding zeros.

$2 x - 1 = - (- 3 x + 3)$

Step 6.4.5.1.3

Apply the distributive property.

$2 x - 1 = - (- 3 x) - 1 \cdot 3$

Step 6.4.5.1.4

Multiply.

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Step 6.4.5.1.4.1

Multiply $- 3$ by $- 1$ .

$2 x - 1 = 3 x - 1 \cdot 3$

Step 6.4.5.1.4.2

Multiply $- 1$ by $3$ .

$2 x - 1 = 3 x - 3$

Step 6.4.5.2

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

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Step 6.4.5.2.1

Subtract $3 x$ from both sides of the equation.

$2 x - 1 - 3 x = - 3$

Step 6.4.5.2.2

Subtract $3 x$ from $2 x$ .

$- x - 1 = - 3$

Step 6.4.5.3

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

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Step 6.4.5.3.1

Add $1$ to both sides of the equation.

$- x = - 3 + 1$

Step 6.4.5.3.2

Add $- 3$ and $1$ .

$- x = - 2$

Step 6.4.5.4

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 6.4.5.4.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 6.4.5.4.2

Simplify the left side.

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Step 6.4.5.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 6.4.5.4.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 6.4.5.4.3

Simplify the right side.

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Step 6.4.5.4.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 6.4.6

Consolidate the solutions.

$x = \frac{4}{5}, 2$

Step 6.5

Solve $x - 2 = - (1 - | 2 x - 1 | - 2 x)$ for $x$ .

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Step 6.5.1

Solve for $| 2 x - 1 |$ .

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Step 6.5.1.1

Rewrite the equation as $- (1 - | 2 x - 1 | - 2 x) = x - 2$ .

$- (1 - | 2 x - 1 | - 2 x) = x - 2$

Step 6.5.1.2

Simplify $- (1 - | 2 x - 1 | - 2 x)$ .

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Step 6.5.1.2.1

Apply the distributive property.

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

Step 6.5.1.2.2

Simplify.

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Step 6.5.1.2.2.1

Multiply $- 1$ by $1$ .

$- 1 - - | 2 x - 1 | - (- 2 x) = x - 2$

Step 6.5.1.2.2.2

Multiply $- - | 2 x - 1 |$ .

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Step 6.5.1.2.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.5.1.2.2.2.2

Multiply $| 2 x - 1 |$ by $1$ .

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

Step 6.5.1.2.2.3

Multiply $- 2$ by $- 1$ .

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

Step 6.5.1.3

Move all terms not containing $| 2 x - 1 |$ to the right side of the equation.

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Step 6.5.1.3.1

Add $1$ to both sides of the equation.

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

Step 6.5.1.3.2

Subtract $2 x$ from both sides of the equation.

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

Step 6.5.1.3.3

Subtract $2 x$ from $x$ .

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

Step 6.5.1.3.4

Add $- 2$ and $1$ .

$| 2 x - 1 | = - x - 1$

Step 6.5.2

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

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

Step 6.5.3

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

$2 x - 1 = - x - 1$

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

Step 6.5.4

Solve $2 x - 1 = - x - 1$ for $x$ .

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Step 6.5.4.1

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

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Step 6.5.4.1.1

Add $x$ to both sides of the equation.

$2 x - 1 + x = - 1$

Step 6.5.4.1.2

Add $2 x$ and $x$ .

$3 x - 1 = - 1$

Step 6.5.4.2

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

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Step 6.5.4.2.1

Add $1$ to both sides of the equation.

$3 x = - 1 + 1$

Step 6.5.4.2.2

Add $- 1$ and $1$ .

$3 x = 0$

Step 6.5.4.3

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

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Step 6.5.4.3.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 6.5.4.3.2

Simplify the left side.

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Step 6.5.4.3.2.1

Cancel the common factor of $3$ .

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Step 6.5.4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 6.5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 6.5.4.3.3

Simplify the right side.

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Step 6.5.4.3.3.1

Divide $0$ by $3$ .

$x = 0$

Step 6.5.5

Solve $2 x - 1 = - (- x - 1)$ for $x$ .

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Step 6.5.5.1

Simplify $- (- x - 1)$ .

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Step 6.5.5.1.1

Rewrite.

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

Step 6.5.5.1.2

Simplify by adding zeros.

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

Step 6.5.5.1.3

Apply the distributive property.

$2 x - 1 = - - x - - 1$

Step 6.5.5.1.4

Multiply $- - x$ .

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Step 6.5.5.1.4.1

Multiply $- 1$ by $- 1$ .

$2 x - 1 = 1 x - - 1$

Step 6.5.5.1.4.2

Multiply $x$ by $1$ .

$2 x - 1 = x - - 1$

Step 6.5.5.1.5

Multiply $- 1$ by $- 1$ .

$2 x - 1 = x + 1$

Step 6.5.5.2

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

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Step 6.5.5.2.1

Subtract $x$ from both sides of the equation.

$2 x - 1 - x = 1$

Step 6.5.5.2.2

Subtract $x$ from $2 x$ .

$x - 1 = 1$

Step 6.5.5.3

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

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Step 6.5.5.3.1

Add $1$ to both sides of the equation.

$x = 1 + 1$

Step 6.5.5.3.2

Add $1$ and $1$ .

$x = 2$

Step 6.5.6

Consolidate the solutions.

$x = 0, 2$

Step 6.6

Consolidate the solutions.

$x = \frac{4}{5}, 2, 0$

Step 7

Consolidate the solutions.

$x = - \frac{2}{5}, 0, \frac{2}{7}, \frac{4}{11}, \frac{4}{5}$

Step 8

Use each root to create test intervals.

$x < - \frac{2}{5}$

$- \frac{2}{5} < x < 0$

$0 < x < \frac{2}{7}$

$\frac{2}{7} < x < \frac{4}{11}$

$\frac{4}{11} < x < \frac{4}{5}$

$x > \frac{4}{5}$

Step 9

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 9.1

Test a value on the interval $x < - \frac{2}{5}$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - \frac{2}{5}$ and see if this value makes the original inequality true.

$x = - 3$

Step 9.1.2

Replace $x$ with $- 3$ in the original inequality.

$| (- 3) - 2 | + | 1 - 5 \cdot - 3 | + | 2 (- 3) - 1 | = 3 (- 3)$

Step 9.1.3

The left side $28$ does not equal to the right side $- 9$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- \frac{2}{5} < x < 0$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- \frac{2}{5} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.2$

Step 9.2.2

Replace $x$ with $- 0.2$ in the original inequality.

$| (- 0.2) - 2 | + | 1 - 5 \cdot - 0.2 | + | 2 (- 0.2) - 1 | = 3 (- 0.2)$

Step 9.2.3

The left side $5.6$ does not equal to the right side $- 0.6$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $0 < x < \frac{2}{7}$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $0 < x < \frac{2}{7}$ and see if this value makes the original inequality true.

$x = 0.14$

Step 9.3.2

Replace $x$ with $0.14$ in the original inequality.

$| (0.14) - 2 | + | 1 - 5 \cdot 0.14 | + | 2 (0.14) - 1 | = 3 (0.14)$

Step 9.3.3

The left side $2.88$ does not equal to the right side $0.42$ , which means that the given statement is false.

False

Step 9.4

Test a value on the interval $\frac{2}{7} < x < \frac{4}{11}$ to see if it makes the inequality true.

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Step 9.4.1

Choose a value on the interval $\frac{2}{7} < x < \frac{4}{11}$ and see if this value makes the original inequality true.

$x = 0.32$

Step 9.4.2

Replace $x$ with $0.32$ in the original inequality.

$| (0.32) - 2 | + | 1 - 5 \cdot 0.32 | + | 2 (0.32) - 1 | = 3 (0.32)$

Step 9.4.3

The left side $2.64$ does not equal to the right side $0.96$ , which means that the given statement is false.

False

Step 9.5

Test a value on the interval $\frac{4}{11} < x < \frac{4}{5}$ to see if it makes the inequality true.

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Step 9.5.1

Choose a value on the interval $\frac{4}{11} < x < \frac{4}{5}$ and see if this value makes the original inequality true.

$x = 0.58$

Step 9.5.2

Replace $x$ with $0.58$ in the original inequality.

$| (0.58) - 2 | + | 1 - 5 \cdot 0.58 | + | 2 (0.58) - 1 | = 3 (0.58)$

Step 9.5.3

The left side $3.48$ does not equal to the right side $1.74$ , which means that the given statement is false.

False

Step 9.6

Test a value on the interval $x > \frac{4}{5}$ to see if it makes the inequality true.

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Step 9.6.1

Choose a value on the interval $x > \frac{4}{5}$ and see if this value makes the original inequality true.

$x = 3$

Step 9.6.2

Replace $x$ with $3$ in the original inequality.

$| (3) - 2 | + | 1 - 5 \cdot 3 | + | 2 (3) - 1 | = 3 (3)$

Step 9.6.3

The left side $20$ does not equal to the right side $9$ , which means that the given statement is false.

False

Step 9.7

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

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 0$ False

$0 < x < \frac{2}{7}$ False

$\frac{2}{7} < x < \frac{4}{11}$ False

$\frac{4}{11} < x < \frac{4}{5}$ False

$x > \frac{4}{5}$ False

$x < - \frac{2}{5}$ False

$- \frac{2}{5} < x < 0$ False

$0 < x < \frac{2}{7}$ False

$\frac{2}{7} < x < \frac{4}{11}$ False

$\frac{4}{11} < x < \frac{4}{5}$ False

$x > \frac{4}{5}$ False

Step 10

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

No solution