Algebra Examples

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

Step 1

Multiply both sides by $x$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $x$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

$| x | = 1 x$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $x$ by $1$ .

$| x | = x$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm x$

Step 3.2

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

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Step 3.2.1

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

$x = x$

Step 3.2.2

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

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Step 3.2.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.2.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.2.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 3.2.4

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

$x = - x$

Step 3.2.5

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

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Step 3.2.5.1

Add $x$ to both sides of the equation.

$x + x = 0$

Step 3.2.5.2

Add $x$ and $x$ .

$2 x = 0$

Step 3.2.6

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 3.2.6.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.6.2

Simplify the left side.

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Step 3.2.6.2.1

Cancel the common factor of $2$ .

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Step 3.2.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 3.2.6.3

Simplify the right side.

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Step 3.2.6.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3.2.7

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

$x =, 0$

Step 4

Exclude the solutions that do not make $\frac{| x |}{x} = 1$ true.

$No solution$