Precalculus Examples

$\frac{1}{2} \cdot | \frac{19}{4} x - \frac{16}{7} | = \frac{8}{7}$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot | \frac{19}{4} x - \frac{16}{7} |) = 2 (\frac{8}{7})$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot | \frac{19}{4} x - \frac{16}{7} |)$ .

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Step 2.1.1.1

Combine $\frac{19}{4}$ and $x$ .

$2 (\frac{1}{2} \cdot | \frac{19 x}{4} - \frac{16}{7} |) = 2 (\frac{8}{7})$

Step 2.1.1.2

Combine $\frac{1}{2}$ and $| \frac{19 x}{4} - \frac{16}{7} |$ .

$2 \frac{| \frac{19 x}{4} - \frac{16}{7} |}{2} = 2 (\frac{8}{7})$

Step 2.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.3.1

Cancel the common factor.

$2 \frac{| \frac{19 x}{4} - \frac{16}{7} |}{2} = 2 (\frac{8}{7})$

Step 2.1.1.3.2

Rewrite the expression.

$| \frac{19 x}{4} - \frac{16}{7} | = 2 (\frac{8}{7})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $2 (\frac{8}{7})$ .

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Step 2.2.1.1

Combine $2$ and $\frac{8}{7}$ .

$| \frac{19 x}{4} - \frac{16}{7} | = \frac{2 \cdot 8}{7}$

Step 2.2.1.2

Multiply $2$ by $8$ .

$| \frac{19 x}{4} - \frac{16}{7} | = \frac{16}{7}$

Step 3

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

$\frac{19 x}{4} - \frac{16}{7} = \pm \frac{16}{7}$

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.

$\frac{19 x}{4} - \frac{16}{7} = \frac{16}{7}$

Step 4.2

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

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Step 4.2.1

Add $\frac{16}{7}$ to both sides of the equation.

$\frac{19 x}{4} = \frac{16}{7} + \frac{16}{7}$

Step 4.2.2

Combine the numerators over the common denominator.

$\frac{19 x}{4} = \frac{16 + 16}{7}$

Step 4.2.3

Add $16$ and $16$ .

$\frac{19 x}{4} = \frac{32}{7}$

Step 4.3

Multiply both sides of the equation by $\frac{4}{19}$ .

$\frac{4}{19} \cdot \frac{19 x}{4} = \frac{4}{19} \cdot \frac{32}{7}$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Simplify $\frac{4}{19} \cdot \frac{19 x}{4}$ .

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Step 4.4.1.1.1

Cancel the common factor of $4$ .

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Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{4}{19} \cdot \frac{19 x}{4} = \frac{4}{19} \cdot \frac{32}{7}$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{19} (19 x) = \frac{4}{19} \cdot \frac{32}{7}$

Step 4.4.1.1.2

Cancel the common factor of $19$ .

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Step 4.4.1.1.2.1

Factor $19$ out of $19 x$ .

$\frac{1}{19} (19 (x)) = \frac{4}{19} \cdot \frac{32}{7}$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{19} (19 x) = \frac{4}{19} \cdot \frac{32}{7}$

Step 4.4.1.1.2.3

Rewrite the expression.

$x = \frac{4}{19} \cdot \frac{32}{7}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Multiply $\frac{4}{19} \cdot \frac{32}{7}$ .

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Step 4.4.2.1.1

Multiply $\frac{4}{19}$ by $\frac{32}{7}$ .

$x = \frac{4 \cdot 32}{19 \cdot 7}$

Step 4.4.2.1.2

Multiply $4$ by $32$ .

$x = \frac{128}{19 \cdot 7}$

Step 4.4.2.1.3

Multiply $19$ by $7$ .

$x = \frac{128}{133}$

Step 4.5

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

$\frac{19 x}{4} - \frac{16}{7} = - \frac{16}{7}$

Step 4.6

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

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Step 4.6.1

Add $\frac{16}{7}$ to both sides of the equation.

$\frac{19 x}{4} = - \frac{16}{7} + \frac{16}{7}$

Step 4.6.2

Combine the numerators over the common denominator.

$\frac{19 x}{4} = \frac{- 16 + 16}{7}$

Step 4.6.3

Add $- 16$ and $16$ .

$\frac{19 x}{4} = \frac{0}{7}$

Step 4.6.4

Divide $0$ by $7$ .

$\frac{19 x}{4} = 0$

Step 4.7

Set the numerator equal to zero.

$19 x = 0$

Step 4.8

Divide each term in $19 x = 0$ by $19$ and simplify.

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Step 4.8.1

Divide each term in $19 x = 0$ by $19$ .

$\frac{19 x}{19} = \frac{0}{19}$

Step 4.8.2

Simplify the left side.

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Step 4.8.2.1

Cancel the common factor of $19$ .

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Step 4.8.2.1.1

Cancel the common factor.

$\frac{19 x}{19} = \frac{0}{19}$

Step 4.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{19}$

Step 4.8.3

Simplify the right side.

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Step 4.8.3.1

Divide $0$ by $19$ .

$x = 0$

Step 4.9

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

$x = \frac{128}{133}, 0$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{128}{133}, 0$

Decimal Form:

$x = 0.96240601 \dots, 0$