Solve the Inequality for x (2(x-4))/x<-4

$\frac{2 (x - 4)}{x} < - 4$

Step 1

Add $4$ to both sides of the inequality.

$\frac{2 (x - 4)}{x} + 4 < 0$

Step 2

Simplify $\frac{2 (x - 4)}{x} + 4$ .

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

$\frac{2 (x - 4)}{x} + \frac{4 x}{x} < 0$

Combine the numerators over the common denominator.

$\frac{2 (x - 4) + 4 x}{x} < 0$

Simplify the numerator.

Factor $2$ out of $2 (x - 4) + 4 x$ .

Factor $2$ out of $4 x$ .

$\frac{2 (x - 4) + 2 (2 x)}{x} < 0$

Factor $2$ out of $2 (x - 4) + 2 (2 x)$ .

$\frac{2 (x - 4 + 2 x)}{x} < 0$

Add $x$ and $2 x$ .

$\frac{2 (3 x - 4)}{x} < 0$

Step 3

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

$x = 0$

$3 x - 4 = 0$

Step 4

Add $4$ to both sides of the equation.

$3 x = 4$

Step 5

Divide each term in $3 x = 4$ by $3$ and simplify.

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 6

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

$x = 0$

$x = \frac{4}{3}$

Step 7

Consolidate the solutions.

$x = 0, \frac{4}{3}$

Step 8

Find the domain of $\frac{2 (3 x - 4)}{x}$ .

Set the denominator in $\frac{2 (3 x - 4)}{x}$ 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 9

Use each root to create test intervals.

$x < 0$

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

$x > \frac{4}{3}$

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 $x < 0$ 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.

$\frac{2 ((- 2) - 4)}{- 2} < - 4$

The left side $6$ is not less than the right side $- 4$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < \frac{4}{3}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{4}{3}$ and see if this value makes the original inequality true.

$x = 1$

Replace $x$ with $1$ in the original inequality.

$\frac{2 ((1) - 4)}{1} < - 4$

The left side $- 6$ is less than the right side $- 4$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{4}{3}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{4}{3}$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{2 ((4) - 4)}{4} < - 4$

The left side $0$ is not less than the right side $- 4$ , which means that the given statement is false.

False

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

$x < 0$ False

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

$x > \frac{4}{3}$ False

$x < 0$ False

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

$x > \frac{4}{3}$ False

Step 11

The solution consists of all of the true intervals.

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

Step 12

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(0, \frac{4}{3})$

Step 13

Solve the Inequality for x (2(x-4))/x<-4

Related Questions

Support LCMGCF.com by disabling your adblocker.