Solve the Inequality for x x>1/x

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

Multiply both sides by $x$ .

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

Simplify.

Simplify the left side.

Multiply $x$ by $x$ .

$x^{2} = \frac{1}{x} x$

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

$x^{2} = \frac{1}{x} x$

Rewrite the expression.

$x^{2} = 1$

Solve for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

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

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

$x = 1$

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

$x = - 1$

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

$x = 1, - 1$

Find the domain of $x - \frac{1}{x}$ .

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 0$

$0 < x < 1$

$x > 1$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$- 4 > \frac{1}{- 4}$

The left side $- 4$ is not greater than the right side $- 0.25$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 0$ and see if this value makes the original inequality true.

$x = - 0.5$

Replace $x$ with $- 0.5$ in the original inequality.

$- 0.5 > \frac{1}{- 0.5}$

The left side $- 0.5$ is greater than the right side $- 2$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Replace $x$ with $0.5$ in the original inequality.

$0.5 > \frac{1}{0.5}$

The left side $0.5$ is not greater than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $x > 1$ to see if it makes the inequality true.

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$4 > \frac{1}{4}$

The left side $4$ is greater than the right side $0.25$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1$ False

$- 1 < x < 0$ True

$0 < x < 1$ False

$x > 1$ True

$x < - 1$ False

$- 1 < x < 0$ True

$0 < x < 1$ False

$x > 1$ True

The solution consists of all of the true intervals.

$- 1 < x < 0$ or $x > 1$

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 0 or x > 1$

Interval Notation:

$(- 1, 0) \cup (1, \infty)$

Solve the Inequality for x x>1/x

Related Questions

Support LCMGCF.com by disabling your adblocker.