Algebra Examples

$2 | 4 x - 1 | = 3 | 4 x + 2 |$

Step 1

Rewrite the equation as $3 | 4 x + 2 | = 2 | 4 x - 1 |$ .

$3 | 4 x + 2 | = 2 | 4 x - 1 |$

Step 2

Divide each term in $3 | 4 x + 2 | = 2 | 4 x - 1 |$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 | 4 x + 2 | = 2 | 4 x - 1 |$ by $3$ .

$\frac{3 | 4 x + 2 |}{3} = \frac{2 | 4 x - 1 |}{3}$

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.

$\frac{3 | 4 x + 2 |}{3} = \frac{2 | 4 x - 1 |}{3}$

Step 2.2.1.2

Divide $| 4 x + 2 |$ by $1$ .

$| 4 x + 2 | = \frac{2 | 4 x - 1 |}{3}$

Step 3

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

$4 x + 2 = \frac{2 (4 x - 1)}{3}$

$4 x + 2 = - \frac{2 (4 x - 1)}{3}$

$- (4 x + 2) = \frac{2 (4 x - 1)}{3}$

$- (4 x + 2) = - \frac{2 (4 x - 1)}{3}$

Step 4

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

$4 x + 2 = \frac{2 (4 x - 1)}{3}$

$4 x + 2 = - \frac{2 (4 x - 1)}{3}$

Step 5

Solve $4 x + 2 = \frac{2 (4 x - 1)}{3}$ for $x$ .

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Step 5.1

Multiply both sides by $3$ .

$(4 x + 2) \cdot 3 = \frac{2 (4 x - 1)}{3} \cdot 3$

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

Simplify $(4 x + 2) \cdot 3$ .

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Step 5.2.1.1.1

Apply the distributive property.

$4 x \cdot 3 + 2 \cdot 3 = \frac{2 (4 x - 1)}{3} \cdot 3$

Step 5.2.1.1.2

Multiply.

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Step 5.2.1.1.2.1

Multiply $3$ by $4$ .

$12 x + 2 \cdot 3 = \frac{2 (4 x - 1)}{3} \cdot 3$

Step 5.2.1.1.2.2

Multiply $2$ by $3$ .

$12 x + 6 = \frac{2 (4 x - 1)}{3} \cdot 3$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{2 (4 x - 1)}{3} \cdot 3$ .

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Step 5.2.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

$12 x + 6 = \frac{2 (4 x - 1)}{3} \cdot 3$

Step 5.2.2.1.1.2

Rewrite the expression.

$12 x + 6 = 2 (4 x - 1)$

Step 5.2.2.1.2

Apply the distributive property.

$12 x + 6 = 2 (4 x) + 2 \cdot - 1$

Step 5.2.2.1.3

Multiply.

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Step 5.2.2.1.3.1

Multiply $4$ by $2$ .

$12 x + 6 = 8 x + 2 \cdot - 1$

Step 5.2.2.1.3.2

Multiply $2$ by $- 1$ .

$12 x + 6 = 8 x - 2$

Step 5.3

Solve for $x$ .

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Step 5.3.1

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

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Step 5.3.1.1

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

$12 x + 6 - 8 x = - 2$

Step 5.3.1.2

Subtract $8 x$ from $12 x$ .

$4 x + 6 = - 2$

Step 5.3.2

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

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Step 5.3.2.1

Subtract $6$ from both sides of the equation.

$4 x = - 2 - 6$

Step 5.3.2.2

Subtract $6$ from $- 2$ .

$4 x = - 8$

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $4$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $x$ by $1$ .

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

Step 5.3.3.3

Simplify the right side.

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Step 5.3.3.3.1

Divide $- 8$ by $4$ .

$x = - 2$

Step 6

Solve $4 x + 2 = - \frac{2 (4 x - 1)}{3}$ for $x$ .

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Step 6.1

Multiply both sides by $3$ .

$(4 x + 2) \cdot 3 = - \frac{2 (4 x - 1)}{3} \cdot 3$

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

Simplify $(4 x + 2) \cdot 3$ .

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Step 6.2.1.1.1

Apply the distributive property.

$4 x \cdot 3 + 2 \cdot 3 = - \frac{2 (4 x - 1)}{3} \cdot 3$

Step 6.2.1.1.2

Multiply.

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Step 6.2.1.1.2.1

Multiply $3$ by $4$ .

$12 x + 2 \cdot 3 = - \frac{2 (4 x - 1)}{3} \cdot 3$

Step 6.2.1.1.2.2

Multiply $2$ by $3$ .

$12 x + 6 = - \frac{2 (4 x - 1)}{3} \cdot 3$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $- \frac{2 (4 x - 1)}{3} \cdot 3$ .

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Step 6.2.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1.1

Move the leading negative in $- \frac{2 (4 x - 1)}{3}$ into the numerator.

$12 x + 6 = \frac{- 2 (4 x - 1)}{3} \cdot 3$

Step 6.2.2.1.1.2

Cancel the common factor.

$12 x + 6 = \frac{- 2 (4 x - 1)}{3} \cdot 3$

Step 6.2.2.1.1.3

Rewrite the expression.

$12 x + 6 = - 2 (4 x - 1)$

Step 6.2.2.1.2

Apply the distributive property.

$12 x + 6 = - 2 (4 x) - 2 \cdot - 1$

Step 6.2.2.1.3

Multiply.

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Step 6.2.2.1.3.1

Multiply $4$ by $- 2$ .

$12 x + 6 = - 8 x - 2 \cdot - 1$

Step 6.2.2.1.3.2

Multiply $- 2$ by $- 1$ .

$12 x + 6 = - 8 x + 2$

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 $8 x$ to both sides of the equation.

$12 x + 6 + 8 x = 2$

Step 6.3.1.2

Add $12 x$ and $8 x$ .

$20 x + 6 = 2$

Step 6.3.2

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

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Step 6.3.2.1

Subtract $6$ from both sides of the equation.

$20 x = 2 - 6$

Step 6.3.2.2

Subtract $6$ from $2$ .

$20 x = - 4$

Step 6.3.3

Divide each term in $20 x = - 4$ by $20$ and simplify.

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Step 6.3.3.1

Divide each term in $20 x = - 4$ by $20$ .

$\frac{20 x}{20} = \frac{- 4}{20}$

Step 6.3.3.2

Simplify the left side.

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Step 6.3.3.2.1

Cancel the common factor of $20$ .

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Step 6.3.3.2.1.1

Cancel the common factor.

$\frac{20 x}{20} = \frac{- 4}{20}$

Step 6.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{20}$

Step 6.3.3.3

Simplify the right side.

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Step 6.3.3.3.1

Cancel the common factor of $- 4$ and $20$ .

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Step 6.3.3.3.1.1

Factor $4$ out of $- 4$ .

$x = \frac{4 (- 1)}{20}$

Step 6.3.3.3.1.2

Cancel the common factors.

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Step 6.3.3.3.1.2.1

Factor $4$ out of $20$ .

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

Step 6.3.3.3.1.2.2

Cancel the common factor.

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

Step 6.3.3.3.1.2.3

Rewrite the expression.

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

Step 6.3.3.3.2

Move the negative in front of the fraction.

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

Step 7

List all of the solutions.

$x = - 2, - \frac{1}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 2, - \frac{1}{5}$

Decimal Form:

$x = - 2, - 0.2$