Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

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.

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

Step 2.2.1.2

Divide $| x |$ by $1$ .

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

Step 3

Rewrite the absolute value equation as four equations without absolute value bars.

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

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

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

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

Step 4

After simplifying, there are only two unique equations to be solved.

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

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

Step 5

Solve $x = \frac{x - 3}{2}$ for $x$ .

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Step 5.1

Multiply both sides by $2$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Move $2$ to the left of $x$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

$2 x = x - 3$

Step 5.3

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

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Step 5.3.1

Subtract $x$ from both sides of the equation.

$2 x - x = - 3$

Step 5.3.2

Subtract $x$ from $2 x$ .

$x = - 3$

Step 6

Solve $x = - \frac{x - 3}{2}$ for $x$ .

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Step 6.1

Multiply both sides by $2$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Move $2$ to the left of $x$ .

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

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $- \frac{x - 3}{2} \cdot 2$ .

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Step 6.2.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.1

Move the leading negative in $- \frac{x - 3}{2}$ into the numerator.

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

Step 6.2.2.1.1.2

Cancel the common factor.

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

Step 6.2.2.1.1.3

Rewrite the expression.

$2 x = - (x - 3)$

Step 6.2.2.1.2

Apply the distributive property.

$2 x = - x - - 3$

Step 6.2.2.1.3

Multiply $- 1$ by $- 3$ .

$2 x = - x + 3$

Step 6.3

Solve for $x$ .

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Step 6.3.1

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

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Step 6.3.1.1

Add $x$ to both sides of the equation.

$2 x + x = 3$

Step 6.3.1.2

Add $2 x$ and $x$ .

$3 x = 3$

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $3$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

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

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $3$ by $3$ .

$x = 1$

Step 7

List all of the solutions.

$x = - 3, 1$