Algebra Examples

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

Step 1

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

$3 x - 4 = 0$

$x + 2 = 0$

Step 2

Add $4$ to both sides of the equation.

$3 x = 4$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $x$ by $1$ .

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

Step 4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

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

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

$x = - 2$

Step 6

Consolidate the solutions.

$x = \frac{4}{3}, - 2$

Step 7

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

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Step 7.1

Set the denominator in $\frac{3 x - 4}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7.3

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

$(- \infty, - 2) \cup (- 2, \infty)$

Step 8

Use each root to create test intervals.

$x < - 2$

$- 2 < x < \frac{4}{3}$

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

$x = - 4$

Step 9.1.2

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

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

Step 9.1.3

The left side $8$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $- 2 < x < \frac{4}{3}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 2 < x < \frac{4}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

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

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

Step 9.2.3

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

False

Step 9.3

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

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Step 9.3.1

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

$x = 4$

Step 9.3.2

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

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

Step 9.3.3

The left side $1.\overline{3}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.4

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

$x < - 2$ True

$- 2 < x < \frac{4}{3}$ False

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

$x < - 2$ True

$- 2 < x < \frac{4}{3}$ False

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

Step 10

The solution consists of all of the true intervals.

$x < - 2$ or $x > \frac{4}{3}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > \frac{4}{3}$

Interval Notation:

$(- \infty, - 2) \cup (\frac{4}{3}, \infty)$

Step 12