Precalculus Examples

$4 | x - 5 | = 10$

Step 1

Divide each term in $4 | x - 5 | = 10$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 | x - 5 | = 10$ by $4$ .

$\frac{4 | x - 5 |}{4} = \frac{10}{4}$

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.

$\frac{4 | x - 5 |}{4} = \frac{10}{4}$

Step 1.2.1.2

Divide $| x - 5 |$ by $1$ .

$| x - 5 | = \frac{10}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $10$ and $4$ .

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Step 1.3.1.1

Factor $2$ out of $10$ .

$| x - 5 | = \frac{2 (5)}{4}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $4$ .

$| x - 5 | = \frac{2 \cdot 5}{2 \cdot 2}$

Step 1.3.1.2.2

Cancel the common factor.

$| x - 5 | = \frac{2 \cdot 5}{2 \cdot 2}$

Step 1.3.1.2.3

Rewrite the expression.

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

Step 2

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

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

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.

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

Step 3.2

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

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Step 3.2.1

Add $5$ to both sides of the equation.

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

Step 3.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.3

Combine $5$ and $\frac{2}{2}$ .

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $5$ by $2$ .

$x = \frac{5 + 10}{2}$

Step 3.2.5.2

Add $5$ and $10$ .

$x = \frac{15}{2}$

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Add $5$ to both sides of the equation.

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

Step 3.4.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.4.3

Combine $5$ and $\frac{2}{2}$ .

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

Step 3.4.4

Combine the numerators over the common denominator.

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

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Multiply $5$ by $2$ .

$x = \frac{- 5 + 10}{2}$

Step 3.4.5.2

Add $- 5$ and $10$ .

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

Step 3.5

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

$x = \frac{15}{2}, \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15}{2}, \frac{5}{2}$

Decimal Form:

$x = 7.5, 2.5$

Mixed Number Form:

$x = 7 \frac{1}{2}, 2 \frac{1}{2}$