Algebra Examples

$3 | x - 1 | + x = - 4 x$

Step 1

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

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Step 1.1

Subtract $x$ from both sides of the equation.

$3 | x - 1 | = - 4 x - x$

Step 1.2

Subtract $x$ from $- 4 x$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

$x - 1 = \pm (- \frac{5 x}{3})$

Step 4

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

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Step 4.1

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

$x - 1 = - \frac{5 x}{3}$

Step 4.2

Multiply both sides by $3$ .

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

Step 4.3

Simplify.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $(x - 1) \cdot 3$ .

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Step 4.3.1.1.1

Apply the distributive property.

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

Step 4.3.1.1.2

Simplify the expression.

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Step 4.3.1.1.2.1

Move $3$ to the left of $x$ .

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

Step 4.3.1.1.2.2

Multiply $- 1$ by $3$ .

$3 x - 3 = - \frac{5 x}{3} \cdot 3$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.1.1

Move the leading negative in $- \frac{5 x}{3}$ into the numerator.

$3 x - 3 = \frac{- 5 x}{3} \cdot 3$

Step 4.3.2.1.2

Cancel the common factor.

$3 x - 3 = \frac{- 5 x}{3} \cdot 3$

Step 4.3.2.1.3

Rewrite the expression.

$3 x - 3 = - 5 x$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

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Step 4.4.1.1

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

$3 x - 3 + 5 x = 0$

Step 4.4.1.2

Add $3 x$ and $5 x$ .

$8 x - 3 = 0$

Step 4.4.2

Add $3$ to both sides of the equation.

$8 x = 3$

Step 4.4.3

Divide each term in $8 x = 3$ by $8$ and simplify.

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Step 4.4.3.1

Divide each term in $8 x = 3$ by $8$ .

$\frac{8 x}{8} = \frac{3}{8}$

Step 4.4.3.2

Simplify the left side.

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Step 4.4.3.2.1

Cancel the common factor of $8$ .

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Step 4.4.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{3}{8}$

Step 4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{8}$

Step 4.5

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

$x - 1 = \frac{5 x}{3}$

Step 4.6

Multiply both sides by $3$ .

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

Step 4.7

Simplify.

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Step 4.7.1

Simplify the left side.

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Step 4.7.1.1

Simplify $(x - 1) \cdot 3$ .

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Step 4.7.1.1.1

Apply the distributive property.

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

Step 4.7.1.1.2

Simplify the expression.

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Step 4.7.1.1.2.1

Move $3$ to the left of $x$ .

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

Step 4.7.1.1.2.2

Multiply $- 1$ by $3$ .

$3 x - 3 = \frac{5 x}{3} \cdot 3$

Step 4.7.2

Simplify the right side.

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Step 4.7.2.1

Cancel the common factor of $3$ .

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Step 4.7.2.1.1

Cancel the common factor.

$3 x - 3 = \frac{5 x}{3} \cdot 3$

Step 4.7.2.1.2

Rewrite the expression.

$3 x - 3 = 5 x$

Step 4.8

Solve for $x$ .

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Step 4.8.1

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

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Step 4.8.1.1

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

$3 x - 3 - 5 x = 0$

Step 4.8.1.2

Subtract $5 x$ from $3 x$ .

$- 2 x - 3 = 0$

Step 4.8.2

Add $3$ to both sides of the equation.

$- 2 x = 3$

Step 4.8.3

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

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Step 4.8.3.1

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

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Step 4.8.3.2

Simplify the left side.

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Step 4.8.3.2.1

Cancel the common factor of $- 2$ .

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Step 4.8.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Step 4.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{- 2}$

Step 4.8.3.3

Simplify the right side.

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Step 4.8.3.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 4.9

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

$x = \frac{3}{8}, - \frac{3}{2}$

Step 5

Exclude the solutions that do not make $3 | x - 1 | + x = - 4 x$ true.

$x = - \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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