Algebra Examples

$\frac{1}{x} - x > 0$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $\frac{1}{x} - x > 0$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{1}{x} - x > 0$ by $x$ .

$\frac{1}{x} x - x \cdot x > 0 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $x$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{1}{x} x - x \cdot x > 0 x$

Step 2.2.1.1.2

Rewrite the expression.

$1 - x \cdot x > 0 x$

Step 2.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.2.1

Move $x$ .

$1 - (x \cdot x) > 0 x$

Step 2.2.1.2.2

Multiply $x$ by $x$ .

$1 - x^{2} > 0 x$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $x$ .

$1 - x^{2} > 0$

Step 3

Solve the inequality.

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Step 3.1

Subtract $1$ from both sides of the inequality.

$- x^{2} > - 1$

Step 3.2

Divide each term in $- x^{2} > - 1$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x^{2} > - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} < \frac{- 1}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} < \frac{- 1}{- 1}$

Step 3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} < \frac{- 1}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} < 1$

Step 3.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} < \sqrt{1}$

Step 3.4

Simplify the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Pull terms out from under the radical.

$| x | < \sqrt{1}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Any root of $1$ is $1$ .

$| x | < 1$

Step 3.5

Write $| x | < 1$ as a piecewise.

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Step 3.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 3.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x < 1$

Step 3.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 3.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x < 1$

Step 3.5.5

Write as a piecewise.

${\begin{cases} x < 1 & x \geq 0 \\ - x < 1 & x < 0 \end{cases}$

Step 3.6

Find the intersection of $x < 1$ and $x \geq 0$ .

$0 \leq x < 1$

Step 3.7

Solve $- x < 1$ when $x < 0$ .

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Step 3.7.1

Divide each term in $- x < 1$ by $- 1$ and simplify.

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Step 3.7.1.1

Divide each term in $- x < 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{1}{- 1}$

Step 3.7.1.2

Simplify the left side.

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Step 3.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{1}{- 1}$

Step 3.7.1.2.2

Divide $x$ by $1$ .

$x > \frac{1}{- 1}$

Step 3.7.1.3

Simplify the right side.

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Step 3.7.1.3.1

Divide $1$ by $- 1$ .

$x > - 1$

Step 3.7.2

Find the intersection of $x > - 1$ and $x < 0$ .

$- 1 < x < 0$

Step 3.8

Find the union of the solutions.

$- 1 < x < 1$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 1$

Interval Notation:

$(- 1, 1)$

Step 5