Solve the Inequality for x (3x-5)/(x+2)<=2

$\frac{3 x - 5}{x + 2} \leq 2$

Step 1

Subtract $2$ from both sides of the inequality.

$\frac{3 x - 5}{x + 2} - 2 \leq 0$

Step 2

Simplify $\frac{3 x - 5}{x + 2} - 2$ .

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

$\frac{3 x - 5}{x + 2} - 2 \cdot \frac{x + 2}{x + 2} \leq 0$

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

$\frac{3 x - 5}{x + 2} + \frac{- 2 (x + 2)}{x + 2} \leq 0$

Combine the numerators over the common denominator.

$\frac{3 x - 5 - 2 (x + 2)}{x + 2} \leq 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x - 5 - 2 x - 2 \cdot 2}{x + 2} \leq 0$

Multiply $- 2$ by $2$ .

$\frac{3 x - 5 - 2 x - 4}{x + 2} \leq 0$

Subtract $2 x$ from $3 x$ .

$\frac{x - 5 - 4}{x + 2} \leq 0$

Subtract $4$ from $- 5$ .

$\frac{x - 9}{x + 2} \leq 0$

Step 3

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

$x - 9 = 0$

$x + 2 = 0$

Step 4

Add $9$ to both sides of the equation.

$x = 9$

Step 5

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

$x = 9$

$x = - 2$

Step 7

Consolidate the solutions.

$x = 9, - 2$

Step 8

Find the domain of $\frac{x - 9}{x + 2}$ .

Set the denominator in $\frac{x - 9}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

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

Step 9

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 9$

$x > 9$

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 $x < - 2$ to see if it makes the inequality true.

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

$x = - 4$

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

$\frac{3 (- 4) - 5}{(- 4) + 2} \leq 2$

The left side $8.5$ is greater than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $- 2 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $- 2 < x < 9$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{3 (0) - 5}{(0) + 2} \leq 2$

The left side $- 2.5$ is less than the right side $2$ , which means that the given statement is always true.

True

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

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

$\frac{3 (12) - 5}{(12) + 2} \leq 2$

The left side $2.2\overline{142857}$ is greater 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.

$x < - 2$ False

$- 2 < x < 9$ True

$x > 9$ False

$x < - 2$ False

$- 2 < x < 9$ True

$x > 9$ False

Step 11

The solution consists of all of the true intervals.

$- 2 < x \leq 9$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x \leq 9$

Interval Notation:

$(- 2, 9]$

Step 13

Solve the Inequality for x (3x-5)/(x+2)<=2

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