Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

$x - 1 = 5 x + 2$

Step 2.2

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

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Step 2.2.1

Subtract $5 x$ from both sides of the equation.

$x - 1 - 5 x = 2$

Step 2.2.2

Subtract $5 x$ from $x$ .

$- 4 x - 1 = 2$

Step 2.3

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

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Step 2.3.1

Add $1$ to both sides of the equation.

$- 4 x = 2 + 1$

Step 2.3.2

Add $2$ and $1$ .

$- 4 x = 3$

Step 2.4

Divide each term in $- 4 x = 3$ by $- 4$ and simplify.

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Step 2.4.1

Divide each term in $- 4 x = 3$ by $- 4$ .

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $- 4$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Divide $x$ by $1$ .

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative in front of the fraction.

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

Step 2.5

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

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

Step 2.6

Simplify $- (5 x + 2)$ .

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Step 2.6.1

Rewrite.

$x - 1 = 0 + 0 - (5 x + 2)$

Step 2.6.2

Simplify by adding zeros.

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

Step 2.6.3

Apply the distributive property.

$x - 1 = - (5 x) - 1 \cdot 2$

Step 2.6.4

Multiply.

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Step 2.6.4.1

Multiply $5$ by $- 1$ .

$x - 1 = - 5 x - 1 \cdot 2$

Step 2.6.4.2

Multiply $- 1$ by $2$ .

$x - 1 = - 5 x - 2$

Step 2.7

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

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Step 2.7.1

Add $5 x$ to both sides of the equation.

$x - 1 + 5 x = - 2$

Step 2.7.2

Add $x$ and $5 x$ .

$6 x - 1 = - 2$

Step 2.8

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

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Step 2.8.1

Add $1$ to both sides of the equation.

$6 x = - 2 + 1$

Step 2.8.2

Add $- 2$ and $1$ .

$6 x = - 1$

Step 2.9

Divide each term in $6 x = - 1$ by $6$ and simplify.

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Step 2.9.1

Divide each term in $6 x = - 1$ by $6$ .

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

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $6$ .

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Step 2.9.2.1.1

Cancel the common factor.

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

Step 2.9.2.1.2

Divide $x$ by $1$ .

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

Step 2.9.3

Simplify the right side.

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Step 2.9.3.1

Move the negative in front of the fraction.

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

Step 2.10

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

$x = - \frac{3}{4}, - \frac{1}{6}$

Step 3

Exclude the solutions that do not make $| x - 1 | = 5 x + 2$ true.

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