Precalculus Examples

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

Step 1

Move all terms not containing $| 4 x - 3 |$ to the right side of the equation.

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Step 1.1

Subtract $2 x$ from both sides of the equation.

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

Step 1.2

Subtract $2 x$ from $4 x$ .

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

Step 2

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

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Step 2.1

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

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

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.

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

Step 2.2.1.2

Divide $| 4 x - 3 |$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

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

Step 2.3.1.1.2

Divide $x$ by $1$ .

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

Step 2.3.1.2

Move the negative in front of the fraction.

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

Step 3

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

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

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.

$4 x - 3 = x - \frac{3}{2}$

Step 4.2

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

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Step 4.2.1

Subtract $x$ from both sides of the equation.

$4 x - 3 - x = - \frac{3}{2}$

Step 4.2.2

Subtract $x$ from $4 x$ .

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

Step 4.3

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

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Step 4.3.1

Add $3$ to both sides of the equation.

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

Step 4.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3.3

Combine $3$ and $\frac{2}{2}$ .

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

Step 4.3.4

Combine the numerators over the common denominator.

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

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $3$ by $2$ .

$3 x = \frac{- 3 + 6}{2}$

Step 4.3.5.2

Add $- 3$ and $6$ .

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

Step 4.4

Divide each term in $3 x = \frac{3}{2}$ by $3$ and simplify.

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Step 4.4.1

Divide each term in $3 x = \frac{3}{2}$ by $3$ .

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

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.1.1

Cancel the common factor.

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

Step 4.4.2.1.2

Divide $x$ by $1$ .

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

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3}{2} \cdot \frac{1}{3}$

Step 4.4.3.2

Cancel the common factor of $3$ .

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Step 4.4.3.2.1

Cancel the common factor.

$x = \frac{3}{2} \cdot \frac{1}{3}$

Step 4.4.3.2.2

Rewrite the expression.

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

Step 4.5

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

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

Step 4.6

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

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Step 4.6.1

Rewrite.

$4 x - 3 = 0 + 0 - (x - \frac{3}{2})$

Step 4.6.2

Simplify by adding zeros.

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

Step 4.6.3

Apply the distributive property.

$4 x - 3 = - x - - \frac{3}{2}$

Step 4.6.4

Multiply $- - \frac{3}{2}$ .

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Step 4.6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.6.4.2

Multiply $\frac{3}{2}$ by $1$ .

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

Step 4.7

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

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Step 4.7.1

Add $x$ to both sides of the equation.

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

Step 4.7.2

Add $4 x$ and $x$ .

$5 x - 3 = \frac{3}{2}$

Step 4.8

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

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Step 4.8.1

Add $3$ to both sides of the equation.

$5 x = \frac{3}{2} + 3$

Step 4.8.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 x = \frac{3}{2} + 3 \cdot \frac{2}{2}$

Step 4.8.3

Combine $3$ and $\frac{2}{2}$ .

$5 x = \frac{3}{2} + \frac{3 \cdot 2}{2}$

Step 4.8.4

Combine the numerators over the common denominator.

$5 x = \frac{3 + 3 \cdot 2}{2}$

Step 4.8.5

Simplify the numerator.

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Step 4.8.5.1

Multiply $3$ by $2$ .

$5 x = \frac{3 + 6}{2}$

Step 4.8.5.2

Add $3$ and $6$ .

$5 x = \frac{9}{2}$

Step 4.9

Divide each term in $5 x = \frac{9}{2}$ by $5$ and simplify.

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Step 4.9.1

Divide each term in $5 x = \frac{9}{2}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{9}{2}}{5}$

Step 4.9.2

Simplify the left side.

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Step 4.9.2.1

Cancel the common factor of $5$ .

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Step 4.9.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{9}{2}}{5}$

Step 4.9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{9}{2}}{5}$

Step 4.9.3

Simplify the right side.

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Step 4.9.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{9}{2} \cdot \frac{1}{5}$

Step 4.9.3.2

Multiply $\frac{9}{2} \cdot \frac{1}{5}$ .

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Step 4.9.3.2.1

Multiply $\frac{9}{2}$ by $\frac{1}{5}$ .

$x = \frac{9}{2 \cdot 5}$

Step 4.9.3.2.2

Multiply $2$ by $5$ .

$x = \frac{9}{10}$

Step 4.10

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

$x = \frac{1}{2}, \frac{9}{10}$

Step 5

Exclude the solutions that do not make $2 | 4 x - 3 | + 2 x = 4 x - 3$ true.

$No solution$