Solve the Inequality for x 1/x>2

$\frac{1}{x} > 2$

Multiply both sides by $x$ .

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

$1 = 2 x$

Solve for $x$ .

Rewrite the equation as $2 x = 1$ .

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Find the domain of $\frac{1}{x} - 2$ .

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.

$x < 0$

$0 < x < \frac{1}{2}$

$x > \frac{1}{2}$

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 $x < 0$ to see if it makes the inequality true.

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

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$\frac{1}{- 2} > 2$

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 $0 < x < \frac{1}{2}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.25$

Replace $x$ with $0.25$ in the original inequality.

$\frac{1}{0.25} > 2$

The left side $4$ is greater than the right side $2$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{1}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$\frac{1}{3} > 2$

The left side $0.\overline{3}$ is not greater than the right side $2$ , which means that the given statement is false.

False

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

$x < 0$ False

$0 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

$x < 0$ False

$0 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

The solution consists of all of the true intervals.

$0 < x < \frac{1}{2}$

The result can be shown in multiple forms.

Inequality Form:

$0 < x < \frac{1}{2}$

Interval Notation:

$(0, \frac{1}{2})$

Solve the Inequality for x 1/x>2

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