Precalculus Examples

$4 | 5 m | = - m + 1$

Step 1

Divide each term in $4 | 5 m | = - m + 1$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 | 5 m | = - m + 1$ by $4$ .

$\frac{4 | 5 m |}{4} = \frac{- m}{4} + \frac{1}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 | 5 m |}{4} = \frac{- m}{4} + \frac{1}{4}$

Step 1.2.1.2

Divide $| 5 m |$ by $1$ .

$| 5 m | = \frac{- m}{4} + \frac{1}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$| 5 m | = - \frac{m}{4} + \frac{1}{4}$

Step 2

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

$5 m = \pm (- \frac{m}{4} + \frac{1}{4})$

Step 3

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

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Step 3.1

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

$5 m = - \frac{m}{4} + \frac{1}{4}$

Step 3.2

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

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Step 3.2.1

Add $\frac{m}{4}$ to both sides of the equation.

$5 m + \frac{m}{4} = \frac{1}{4}$

Step 3.2.2

To write $5 m$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 m \cdot \frac{4}{4} + \frac{m}{4} = \frac{1}{4}$

Step 3.2.3

Combine $5 m$ and $\frac{4}{4}$ .

$\frac{5 m \cdot 4}{4} + \frac{m}{4} = \frac{1}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{5 m \cdot 4 + m}{4} = \frac{1}{4}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Factor $m$ out of $5 m \cdot 4 + m$ .

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Step 3.2.5.1.1

Factor $m$ out of $5 m \cdot 4$ .

$\frac{m (5 \cdot 4) + m}{4} = \frac{1}{4}$

Step 3.2.5.1.2

Raise $m$ to the power of $1$ .

$\frac{m (5 \cdot 4) + m^{1}}{4} = \frac{1}{4}$

Step 3.2.5.1.3

Factor $m$ out of $m^{1}$ .

$\frac{m (5 \cdot 4) + m \cdot 1}{4} = \frac{1}{4}$

Step 3.2.5.1.4

Factor $m$ out of $m (5 \cdot 4) + m \cdot 1$ .

$\frac{m (5 \cdot 4 + 1)}{4} = \frac{1}{4}$

Step 3.2.5.2

Multiply $5$ by $4$ .

$\frac{m (20 + 1)}{4} = \frac{1}{4}$

Step 3.2.5.3

Add $20$ and $1$ .

$\frac{m \cdot 21}{4} = \frac{1}{4}$

Step 3.2.6

Move $21$ to the left of $m$ .

$\frac{21 m}{4} = \frac{1}{4}$

Step 3.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$21 m = 1$

Step 3.4

Divide each term in $21 m = 1$ by $21$ and simplify.

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Step 3.4.1

Divide each term in $21 m = 1$ by $21$ .

$\frac{21 m}{21} = \frac{1}{21}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $21$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{21 m}{21} = \frac{1}{21}$

Step 3.4.2.1.2

Divide $m$ by $1$ .

$m = \frac{1}{21}$

Step 3.5

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

$5 m = - (- \frac{m}{4} + \frac{1}{4})$

Step 3.6

Simplify $- (- \frac{m}{4} + \frac{1}{4})$ .

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Step 3.6.1

Apply the distributive property.

$5 m = - - \frac{m}{4} - \frac{1}{4}$

Step 3.6.2

Multiply $- - \frac{m}{4}$ .

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Step 3.6.2.1

Multiply $- 1$ by $- 1$ .

$5 m = 1 \frac{m}{4} - \frac{1}{4}$

Step 3.6.2.2

Multiply $\frac{m}{4}$ by $1$ .

$5 m = \frac{m}{4} - \frac{1}{4}$

Step 3.7

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

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Step 3.7.1

Subtract $\frac{m}{4}$ from both sides of the equation.

$5 m - \frac{m}{4} = - \frac{1}{4}$

Step 3.7.2

To write $5 m$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.7.3

Combine $5 m$ and $\frac{4}{4}$ .

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

Step 3.7.4

Combine the numerators over the common denominator.

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

Step 3.7.5

Simplify the numerator.

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Step 3.7.5.1

Factor $m$ out of $5 m \cdot 4 - m$ .

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Step 3.7.5.1.1

Factor $m$ out of $5 m \cdot 4$ .

$\frac{m (5 \cdot 4) - m}{4} = - \frac{1}{4}$

Step 3.7.5.1.2

Factor $m$ out of $- m$ .

$\frac{m (5 \cdot 4) + m \cdot - 1}{4} = - \frac{1}{4}$

Step 3.7.5.1.3

Factor $m$ out of $m (5 \cdot 4) + m \cdot - 1$ .

$\frac{m (5 \cdot 4 - 1)}{4} = - \frac{1}{4}$

Step 3.7.5.2

Multiply $5$ by $4$ .

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

Step 3.7.5.3

Subtract $1$ from $20$ .

$\frac{m \cdot 19}{4} = - \frac{1}{4}$

Step 3.7.6

Move $19$ to the left of $m$ .

$\frac{19 m}{4} = - \frac{1}{4}$

Step 3.8

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$19 m = - 1$

Step 3.9

Divide each term in $19 m = - 1$ by $19$ and simplify.

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Step 3.9.1

Divide each term in $19 m = - 1$ by $19$ .

$\frac{19 m}{19} = \frac{- 1}{19}$

Step 3.9.2

Simplify the left side.

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Step 3.9.2.1

Cancel the common factor of $19$ .

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Step 3.9.2.1.1

Cancel the common factor.

$\frac{19 m}{19} = \frac{- 1}{19}$

Step 3.9.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 1}{19}$

Step 3.9.3

Simplify the right side.

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Step 3.9.3.1

Move the negative in front of the fraction.

$m = - \frac{1}{19}$

Step 3.10

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

$m = \frac{1}{21}, - \frac{1}{19}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$m = \frac{1}{21}, - \frac{1}{19}$

Decimal Form:

$m = 0.\overline{047619}, - 0.05263157 \dots$