Solve the Inequality for x 1/x<1/(x-3)

$\frac{1}{x} < \frac{1}{x - 3}$

Step 1

Subtract $\frac{1}{x - 3}$ from both sides of the inequality.

$\frac{1}{x} - \frac{1}{x - 3} < 0$

Step 2

Simplify $\frac{1}{x} - \frac{1}{x - 3}$ .

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{1}{x} \cdot \frac{x - 3}{x - 3} - \frac{1}{x - 3} < 0$

To write $- \frac{1}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} \cdot \frac{x - 3}{x - 3} - \frac{1}{x - 3} \cdot \frac{x}{x} < 0$

Write each expression with a common denominator of $x (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x}$ by $\frac{x - 3}{x - 3}$ .

$\frac{x - 3}{x (x - 3)} - \frac{1}{x - 3} \cdot \frac{x}{x} < 0$

Multiply $\frac{1}{x - 3}$ by $\frac{x}{x}$ .

$\frac{x - 3}{x (x - 3)} - \frac{x}{(x - 3) x} < 0$

Reorder the factors of $(x - 3) x$ .

$\frac{x - 3}{x (x - 3)} - \frac{x}{x (x - 3)} < 0$

Combine the numerators over the common denominator.

$\frac{x - 3 - x}{x (x - 3)} < 0$

Simplify the numerator.

Subtract $x$ from $x$ .

$\frac{0 - 3}{x (x - 3)} < 0$

Subtract $3$ from $0$ .

$\frac{- 3}{x (x - 3)} < 0$

Move the negative in front of the fraction.

$- \frac{3}{x (x - 3)} < 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x = 0$

$x - 3 = 0$

Step 4

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

$x = 3$

Step 6

Consolidate the solutions.

$x = 0, 3$

Step 7

Find the domain of $- \frac{3}{x (x - 3)}$ .

Set the denominator in $\frac{3}{x (x - 3)}$ equal to $0$ to find where the expression is undefined.

$x (x - 3) = 0$

Solve for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

The final solution is all the values that make $x (x - 3) = 0$ true.

$x = 0, 3$

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

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

Step 8

Use each root to create test intervals.

$x < 0$

$0 < x < 3$

$x > 3$

Step 9

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.

$\frac{1}{- 2} < \frac{1}{(- 2) - 3}$

The left side $- 0.5$ is less than the right side $- 0.2$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

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

$\frac{1}{2} < \frac{1}{(2) - 3}$

The left side $0.5$ is not less than the right side $- 1$ , which means that the given statement is false.

False

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$\frac{1}{6} < \frac{1}{(6) - 3}$

The left side $0.1\overline{6}$ is less than the right side $0.\overline{3}$ , which means that the given statement is always true.

True

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

$x < 0$ True

$0 < x < 3$ False

$x > 3$ True

$x < 0$ True

$0 < x < 3$ False

$x > 3$ True

Step 10

The solution consists of all of the true intervals.

$x < 0$ or $x > 3$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > 3$

Interval Notation:

$(- \infty, 0) \cup (3, \infty)$

Step 12

Solve the Inequality for x 1/x<1/(x-3)

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