Calculus Examples

$| 4 x - 5 | = 10$

Step 1

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

$4 x - 5 = \pm 10$

Step 2

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

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Step 2.1

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

$4 x - 5 = 10$

Step 2.2

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

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Step 2.2.1

Add $5$ to both sides of the equation.

$4 x = 10 + 5$

Step 2.2.2

Add $10$ and $5$ .

$4 x = 15$

Step 2.3

Divide each term in $4 x = 15$ by $4$ and simplify.

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Step 2.3.1

Divide each term in $4 x = 15$ by $4$ .

$\frac{4 x}{4} = \frac{15}{4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{15}{4}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{4}$

Step 2.4

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

$4 x - 5 = - 10$

Step 2.5

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

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Step 2.5.1

Add $5$ to both sides of the equation.

$4 x = - 10 + 5$

Step 2.5.2

Add $- 10$ and $5$ .

$4 x = - 5$

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $4$ .

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Step 2.6.2.1.1

Cancel the common factor.

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

Step 2.6.2.1.2

Divide $x$ by $1$ .

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

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Move the negative in front of the fraction.

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

Step 2.7

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

$x = \frac{15}{4}, - \frac{5}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15}{4}, - \frac{5}{4}$

Decimal Form:

$x = 3.75, - 1.25$

Mixed Number Form:

$x = 3 \frac{3}{4}, - 1 \frac{1}{4}$