Algebra Examples

$- 2 | x - 1 | + 3 = \frac{2}{5} x - \frac{11}{5}$

Step 1

Combine $\frac{2}{5}$ and $x$ .

$- 2 | x - 1 | + 3 = \frac{2 x}{5} - \frac{11}{5}$

Step 2

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

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Step 2.1

Subtract $3$ from both sides of the equation.

$- 2 | x - 1 | = \frac{2 x}{5} - \frac{11}{5} - 3$

Step 2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 2 | x - 1 | = \frac{2 x}{5} - \frac{11}{5} - 3 \cdot \frac{5}{5}$

Step 2.3

Combine $- 3$ and $\frac{5}{5}$ .

$- 2 | x - 1 | = \frac{2 x}{5} - \frac{11}{5} + \frac{- 3 \cdot 5}{5}$

Step 2.4

Combine the numerators over the common denominator.

$- 2 | x - 1 | = \frac{2 x}{5} + \frac{- 11 - 3 \cdot 5}{5}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 3$ by $5$ .

$- 2 | x - 1 | = \frac{2 x}{5} + \frac{- 11 - 15}{5}$

Step 2.5.2

Subtract $15$ from $- 11$ .

$- 2 | x - 1 | = \frac{2 x}{5} + \frac{- 26}{5}$

Step 2.6

Move the negative in front of the fraction.

$- 2 | x - 1 | = \frac{2 x}{5} - \frac{26}{5}$

Step 3

Divide each term in $- 2 | x - 1 | = \frac{2 x}{5} - \frac{26}{5}$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 | x - 1 | = \frac{2 x}{5} - \frac{26}{5}$ by $- 2$ .

$\frac{- 2 | x - 1 |}{- 2} = \frac{\frac{2 x}{5}}{- 2} + \frac{- \frac{26}{5}}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 | x - 1 |}{- 2} = \frac{\frac{2 x}{5}}{- 2} + \frac{- \frac{26}{5}}{- 2}$

Step 3.2.1.2

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

$| x - 1 | = \frac{\frac{2 x}{5}}{- 2} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$| x - 1 | = \frac{2 x}{5} \cdot \frac{1}{- 2} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.2.1

Factor $2$ out of $2 x$ .

$| x - 1 | = \frac{2 (x)}{5} \cdot \frac{1}{- 2} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.2.2

Factor $2$ out of $- 2$ .

$| x - 1 | = \frac{2 (x)}{5} \cdot \frac{1}{2 (- 1)} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.2.3

Cancel the common factor.

$| x - 1 | = \frac{2 x}{5} \cdot \frac{1}{2 \cdot - 1} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.2.4

Rewrite the expression.

$| x - 1 | = \frac{x}{5} \cdot \frac{1}{- 1} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.3

Multiply $\frac{x}{5}$ by $\frac{1}{- 1}$ .

$| x - 1 | = \frac{x}{5 \cdot - 1} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.4

Multiply $5$ by $- 1$ .

$| x - 1 | = \frac{x}{- 5} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.5

Move the negative in front of the fraction.

$| x - 1 | = - \frac{x}{5} + \frac{- \frac{26}{5}}{- 2}$

Step 3.3.1.6

Multiply the numerator by the reciprocal of the denominator.

$| x - 1 | = - \frac{x}{5} - \frac{26}{5} \cdot \frac{1}{- 2}$

Step 3.3.1.7

Cancel the common factor of $2$ .

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Step 3.3.1.7.1

Move the leading negative in $- \frac{26}{5}$ into the numerator.

$| x - 1 | = - \frac{x}{5} + \frac{- 26}{5} \cdot \frac{1}{- 2}$

Step 3.3.1.7.2

Factor $2$ out of $- 26$ .

$| x - 1 | = - \frac{x}{5} + \frac{2 (- 13)}{5} \cdot \frac{1}{- 2}$

Step 3.3.1.7.3

Factor $2$ out of $- 2$ .

$| x - 1 | = - \frac{x}{5} + \frac{2 \cdot - 13}{5} \cdot \frac{1}{2 \cdot - 1}$

Step 3.3.1.7.4

Cancel the common factor.

$| x - 1 | = - \frac{x}{5} + \frac{2 \cdot - 13}{5} \cdot \frac{1}{2 \cdot - 1}$

Step 3.3.1.7.5

Rewrite the expression.

$| x - 1 | = - \frac{x}{5} + \frac{- 13}{5} \cdot \frac{1}{- 1}$

Step 3.3.1.8

Multiply $\frac{- 13}{5}$ by $\frac{1}{- 1}$ .

$| x - 1 | = - \frac{x}{5} + \frac{- 13}{5 \cdot - 1}$

Step 3.3.1.9

Multiply $5$ by $- 1$ .

$| x - 1 | = - \frac{x}{5} + \frac{- 13}{- 5}$

Step 3.3.1.10

Dividing two negative values results in a positive value.

$| x - 1 | = - \frac{x}{5} + \frac{13}{5}$

Step 4

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

$x - 1 = \pm (- \frac{x}{5} + \frac{13}{5})$

Step 5

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

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 1 = - \frac{x}{5} + \frac{13}{5}$

Step 5.2

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

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Step 5.2.1

Add $\frac{x}{5}$ to both sides of the equation.

$x - 1 + \frac{x}{5} = \frac{13}{5}$

Step 5.2.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x \cdot \frac{5}{5} + \frac{x}{5} - 1 = \frac{13}{5}$

Step 5.2.3

Combine $x$ and $\frac{5}{5}$ .

$\frac{x \cdot 5}{5} + \frac{x}{5} - 1 = \frac{13}{5}$

Step 5.2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 5 + x}{5} - 1 = \frac{13}{5}$

Step 5.2.5

Add $x \cdot 5$ and $x$ .

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Step 5.2.5.1

Reorder $x$ and $5$ .

$\frac{5 \cdot x + x}{5} - 1 = \frac{13}{5}$

Step 5.2.5.2

Add $5 \cdot x$ and $x$ .

$\frac{6 \cdot x}{5} - 1 = \frac{13}{5}$

$\frac{6 x}{5} - 1 = \frac{13}{5}$

Step 5.3

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

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Step 5.3.1

Add $1$ to both sides of the equation.

$\frac{6 x}{5} = \frac{13}{5} + 1$

Step 5.3.2

Write $1$ as a fraction with a common denominator.

$\frac{6 x}{5} = \frac{13}{5} + \frac{5}{5}$

Step 5.3.3

Combine the numerators over the common denominator.

$\frac{6 x}{5} = \frac{13 + 5}{5}$

Step 5.3.4

Add $13$ and $5$ .

$\frac{6 x}{5} = \frac{18}{5}$

Step 5.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$6 x = 18$

Step 5.5

Divide each term in $6 x = 18$ by $6$ and simplify.

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Step 5.5.1

Divide each term in $6 x = 18$ by $6$ .

$\frac{6 x}{6} = \frac{18}{6}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $6$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{18}{6}$

Step 5.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{6}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $18$ by $6$ .

$x = 3$

Step 5.6

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

$x - 1 = - (- \frac{x}{5} + \frac{13}{5})$

Step 5.7

Simplify $- (- \frac{x}{5} + \frac{13}{5})$ .

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Step 5.7.1

Rewrite.

$x - 1 = 0 + 0 - (- \frac{x}{5} + \frac{13}{5})$

Step 5.7.2

Simplify by adding zeros.

$x - 1 = - (- \frac{x}{5} + \frac{13}{5})$

Step 5.7.3

Apply the distributive property.

$x - 1 = - - \frac{x}{5} - \frac{13}{5}$

Step 5.7.4

Multiply $- - \frac{x}{5}$ .

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Step 5.7.4.1

Multiply $- 1$ by $- 1$ .

$x - 1 = 1 \frac{x}{5} - \frac{13}{5}$

Step 5.7.4.2

Multiply $\frac{x}{5}$ by $1$ .

$x - 1 = \frac{x}{5} - \frac{13}{5}$

Step 5.8

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

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Step 5.8.1

Subtract $\frac{x}{5}$ from both sides of the equation.

$x - 1 - \frac{x}{5} = - \frac{13}{5}$

Step 5.8.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x \cdot \frac{5}{5} - \frac{x}{5} - 1 = - \frac{13}{5}$

Step 5.8.3

Combine $x$ and $\frac{5}{5}$ .

$\frac{x \cdot 5}{5} - \frac{x}{5} - 1 = - \frac{13}{5}$

Step 5.8.4

Combine the numerators over the common denominator.

$\frac{x \cdot 5 - x}{5} - 1 = - \frac{13}{5}$

Step 5.8.5

Subtract $x$ from $x \cdot 5$ .

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Step 5.8.5.1

Reorder $x$ and $5$ .

$\frac{5 \cdot x - x}{5} - 1 = - \frac{13}{5}$

Step 5.8.5.2

Subtract $x$ from $5 \cdot x$ .

$\frac{4 \cdot x}{5} - 1 = - \frac{13}{5}$

$\frac{4 x}{5} - 1 = - \frac{13}{5}$

Step 5.9

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

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Step 5.9.1

Add $1$ to both sides of the equation.

$\frac{4 x}{5} = - \frac{13}{5} + 1$

Step 5.9.2

Write $1$ as a fraction with a common denominator.

$\frac{4 x}{5} = - \frac{13}{5} + \frac{5}{5}$

Step 5.9.3

Combine the numerators over the common denominator.

$\frac{4 x}{5} = \frac{- 13 + 5}{5}$

Step 5.9.4

Add $- 13$ and $5$ .

$\frac{4 x}{5} = \frac{- 8}{5}$

Step 5.9.5

Move the negative in front of the fraction.

$\frac{4 x}{5} = - \frac{8}{5}$

Step 5.10

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 x = - 8$

Step 5.11

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

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Step 5.11.1

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

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

Step 5.11.2

Simplify the left side.

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Step 5.11.2.1

Cancel the common factor of $4$ .

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Step 5.11.2.1.1

Cancel the common factor.

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

Step 5.11.2.1.2

Divide $x$ by $1$ .

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

Step 5.11.3

Simplify the right side.

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Step 5.11.3.1

Divide $- 8$ by $4$ .

$x = - 2$

Step 5.12

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

$x = 3, - 2$