Solve the Inequality for x (3x-2)/(x+4)<2

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

Step 1

Subtract $2$ from both sides of the inequality.

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

Step 2

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

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Combine $- 2$ and $\frac{x + 4}{x + 4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 2$ by $4$ .

$\frac{3 x - 2 - 2 x - 8}{x + 4} < 0$

Subtract $2 x$ from $3 x$ .

$\frac{x - 2 - 8}{x + 4} < 0$

Subtract $8$ from $- 2$ .

$\frac{x - 10}{x + 4} < 0$

Step 3

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

$x - 10 = 0$

$x + 4 = 0$

Step 4

Add $10$ to both sides of the equation.

$x = 10$

Step 5

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6

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

$x = 10$

$x = - 4$

Step 7

Consolidate the solutions.

$x = 10, - 4$

Step 8

Find the domain of $\frac{x - 10}{x + 4}$ .

Set the denominator in $\frac{x - 10}{x + 4}$ equal to $0$ to find where the expression is undefined.

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

Step 9

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 10$

$x > 10$

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 $x < - 4$ 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.

$\frac{3 (- 6) - 2}{(- 6) + 4} < 2$

The left side $10$ is not less than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 10$ and see if this value makes the original inequality true.

$x = 0$

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

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

The left side $- 0.5$ is less than the right side $2$ , which means that the given statement is always true.

True

Test a value on the interval $x > 10$ to see if it makes the inequality true.

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$\frac{3 (12) - 2}{(12) + 4} < 2$

The left side $2.125$ is not less than the right side $2$ , which means that the given statement is false.

False

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

$x < - 4$ False

$- 4 < x < 10$ True

$x > 10$ False

$x < - 4$ False

$- 4 < x < 10$ True

$x > 10$ False

Step 11

The solution consists of all of the true intervals.

$- 4 < x < 10$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < 10$

Interval Notation:

$(- 4, 10)$

Step 13

Solve the Inequality for x (3x-2)/(x+4)<2

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