Solve the Inequality for x x/(x-3)<4

$\frac{x}{x - 3} < 4$

Step 1

Subtract $4$ from both sides of the inequality.

$\frac{x}{x - 3} - 4 < 0$

Step 2

Simplify $\frac{x}{x - 3} - 4$ .

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x}{x - 3} - 4 \cdot \frac{x - 3}{x - 3} < 0$

Combine $- 4$ and $\frac{x - 3}{x - 3}$ .

$\frac{x}{x - 3} + \frac{- 4 (x - 3)}{x - 3} < 0$

Combine the numerators over the common denominator.

$\frac{x - 4 (x - 3)}{x - 3} < 0$

Simplify the numerator.

Apply the distributive property.

$\frac{x - 4 x - 4 \cdot - 3}{x - 3} < 0$

Multiply $- 4$ by $- 3$ .

$\frac{x - 4 x + 12}{x - 3} < 0$

Subtract $4 x$ from $x$ .

$\frac{- 3 x + 12}{x - 3} < 0$

Factor $3$ out of $- 3 x + 12$ .

Factor $3$ out of $- 3 x$ .

$\frac{3 (- x) + 12}{x - 3} < 0$

Factor $3$ out of $12$ .

$\frac{3 (- x) + 3 (4)}{x - 3} < 0$

Factor $3$ out of $3 (- x) + 3 (4)$ .

$\frac{3 (- x + 4)}{x - 3} < 0$

Factor $- 1$ out of $- x$ .

$\frac{3 (- (x) + 4)}{x - 3} < 0$

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{3 (- (x) - 1 (- 4))}{x - 3} < 0$

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{3 (- (x - 4))}{x - 3} < 0$

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{3 (- 1 (x - 4))}{x - 3} < 0$

Move the negative in front of the fraction.

$- \frac{3 (x - 4)}{x - 3} < 0$

Step 3

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

$x - 4 = 0$

$x - 3 = 0$

Step 4

Add $4$ to both sides of the equation.

$x = 4$

Step 5

Add $3$ to both sides of the equation.

$x = 3$

Step 6

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

$x = 4$

$x = 3$

Step 7

Consolidate the solutions.

$x = 4, 3$

Step 8

Find the domain of $- \frac{3 (x - 4)}{x - 3}$ .

Set the denominator in $\frac{3 (x - 4)}{x - 3}$ equal to $0$ to find where the expression is undefined.

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

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

Step 9

Use each root to create test intervals.

$x < 3$

$3 < x < 4$

$x > 4$

Step 10

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 $x < 3$ 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.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$\frac{0}{(0) - 3} < 4$

The left side $0$ is less than the right side $4$ , which means that the given statement is always true.

True

Test a value on the interval $3 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $3 < x < 4$ and see if this value makes the original inequality true.

$x = 3.5$

Replace $x$ with $3.5$ in the original inequality.

$\frac{3.5}{(3.5) - 3} < 4$

The left side $7$ is not less than the right side $4$ , which means that the given statement is false.

False

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

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

$\frac{6}{(6) - 3} < 4$

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

True

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

$x < 3$ True

$3 < x < 4$ False

$x > 4$ True

$x < 3$ True

$3 < x < 4$ False

$x > 4$ True

Step 11

The solution consists of all of the true intervals.

$x < 3$ or $x > 4$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < 3 or x > 4$

Interval Notation:

$(- \infty, 3) \cup (4, \infty)$

Step 13

Solve the Inequality for x x/(x-3)<4

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