Solve the Inequality for x 2/x<5

$\frac{2}{x} < 5$

Step 1

Multiply both sides by $x$ .

$\frac{2}{x} x = 5 x$

Step 2

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{2}{x} x = 5 x$

Rewrite the expression.

$2 = 5 x$

Step 3

Solve for $x$ .

Rewrite the equation as $5 x = 2$ .

$5 x = 2$

Divide each term in $5 x = 2$ by $5$ and simplify.

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 4

Find the domain of $\frac{2}{x} - 5$ .

Set the denominator in $\frac{2}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

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

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

Step 5

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{2}{5}$

$x > \frac{2}{5}$

Step 6

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.

$\frac{2}{- 2} < 5$

The left side $- 1$ is less than the right side $5$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < \frac{2}{5}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{2}{5}$ and see if this value makes the original inequality true.

$x = 0.2$

Replace $x$ with $0.2$ in the original inequality.

$\frac{2}{0.2} < 5$

The left side $10$ is not less than the right side $5$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{2}{5}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{2}{5}$ and see if this value makes the original inequality true.

$x = 3$

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

$\frac{2}{3} < 5$

The left side $0.\overline{6}$ is less than the right side $5$ , which means that the given statement is always true.

True

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

$x < 0$ True

$0 < x < \frac{2}{5}$ False

$x > \frac{2}{5}$ True

$x < 0$ True

$0 < x < \frac{2}{5}$ False

$x > \frac{2}{5}$ True

Step 7

The solution consists of all of the true intervals.

$x < 0$ or $x > \frac{2}{5}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > \frac{2}{5}$

Interval Notation:

$(- \infty, 0) \cup (\frac{2}{5}, \infty)$

Step 9

Solve the Inequality for x 2/x<5

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