Pre-Algebra Examples

$\frac{3 x - 2}{x} - 4 > 0$

Step 1

Simplify $\frac{3 x - 2}{x} - 4$ .

Tap for more steps...

Step 1.1

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

$\frac{3 x - 2}{x} - 4 \cdot \frac{x}{x} > 0$

Step 1.2

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

$\frac{3 x - 2}{x} + \frac{- 4 x}{x} > 0$

Step 1.3

Combine the numerators over the common denominator.

$\frac{3 x - 2 - 4 x}{x} > 0$

Step 1.4

Subtract $4 x$ from $3 x$ .

$\frac{- x - 2}{x} > 0$

Step 1.5

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

$\frac{- (x) - 2}{x} > 0$

Step 1.6

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (x) - 1 (2)}{x} > 0$

Step 1.7

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 1.8

Simplify the expression.

Tap for more steps...

Step 1.8.1

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$\frac{- 1 (x + 2)}{x} > 0$

Step 1.8.2

Move the negative in front of the fraction.

$- \frac{x + 2}{x} > 0$

Step 2

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

$x = 0$

$x + 2 = 0$

Step 3

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

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

$x = 0$

$x = - 2$

Step 5

Consolidate the solutions.

$x = 0, - 2$

Step 6

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

Tap for more steps...

Step 6.1

Set the denominator in $\frac{3 x - 2}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 6.2

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

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

Step 7

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$x > 0$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 8.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 8.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 8.1.2

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

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

Step 8.1.3

The left side $- 0.5$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 8.2.1

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 8.2.2

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

$\frac{3 (- 1) - 2}{- 1} - 4 > 0$

Step 8.2.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.3

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

Tap for more steps...

Step 8.3.1

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

$x = 2$

Step 8.3.2

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

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

Step 8.3.3

The left side $- 2$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 8.4

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

$x < - 2$ False

$- 2 < x < 0$ True

$x > 0$ False

$x < - 2$ False

$- 2 < x < 0$ True

$x > 0$ False

Step 9

The solution consists of all of the true intervals.

$- 2 < x < 0$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 0$

Interval Notation:

$(- 2, 0)$

Step 11