Solve the Inequality for x (3x+1)/(x-2)>=5

$\frac{3 x + 1}{x - 2} \geq 5$

Step 1

Subtract $5$ from both sides of the inequality.

$\frac{3 x + 1}{x - 2} - 5 \geq 0$

Step 2

Simplify $\frac{3 x + 1}{x - 2} - 5$ .

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{3 x + 1}{x - 2} - 5 \cdot \frac{x - 2}{x - 2} \geq 0$

Combine $- 5$ and $\frac{x - 2}{x - 2}$ .

$\frac{3 x + 1}{x - 2} + \frac{- 5 (x - 2)}{x - 2} \geq 0$

Combine the numerators over the common denominator.

$\frac{3 x + 1 - 5 (x - 2)}{x - 2} \geq 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 1 - 5 x - 5 \cdot - 2}{x - 2} \geq 0$

Multiply $- 5$ by $- 2$ .

$\frac{3 x + 1 - 5 x + 10}{x - 2} \geq 0$

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x + 1 + 10}{x - 2} \geq 0$

Add $1$ and $10$ .

$\frac{- 2 x + 11}{x - 2} \geq 0$

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

$\frac{- (2 x) + 11}{x - 2} \geq 0$

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{- (2 x) - 1 (- 11)}{x - 2} \geq 0$

Factor $- 1$ out of $- (2 x) - 1 (- 11)$ .

$\frac{- (2 x - 11)}{x - 2} \geq 0$

Rewrite $- (2 x - 11)$ as $- 1 (2 x - 11)$ .

$\frac{- 1 (2 x - 11)}{x - 2} \geq 0$

Move the negative in front of the fraction.

$- \frac{2 x - 11}{x - 2} \geq 0$

Step 3

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

$2 x - 11 = 0$

$x - 2 = 0$

Step 4

Add $11$ to both sides of the equation.

$2 x = 11$

Step 5

Divide each term in $2 x = 11$ by $2$ and simplify.

Divide each term in $2 x = 11$ by $2$ .

$\frac{2 x}{2} = \frac{11}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{11}{2}$

Divide $x$ by $1$ .

$x = \frac{11}{2}$

Step 6

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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

$x = \frac{11}{2}$

$x = 2$

Step 8

Consolidate the solutions.

$x = \frac{11}{2}, 2$

Step 9

Find the domain of $- \frac{2 x - 11}{x - 2}$ .

Set the denominator in $\frac{2 x - 11}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Add $2$ to 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 10

Use each root to create test intervals.

$x < 2$

$2 < x < \frac{11}{2}$

$x > \frac{11}{2}$

Step 11

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.

$x = 0$

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

$\frac{3 (0) + 1}{(0) - 2} \geq 5$

The left side $- 0.5$ is less than the right side $5$ , which means that the given statement is false.

False

Test a value on the interval $2 < x < \frac{11}{2}$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < \frac{11}{2}$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{3 (4) + 1}{(4) - 2} \geq 5$

The left side $6.5$ is greater than the right side $5$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{11}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{11}{2}$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$\frac{3 (8) + 1}{(8) - 2} \geq 5$

The left side $4.1\overline{6}$ is less than the right side $5$ , which means that the given statement is false.

False

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

$x < 2$ False

$2 < x < \frac{11}{2}$ True

$x > \frac{11}{2}$ False

$x < 2$ False

$2 < x < \frac{11}{2}$ True

$x > \frac{11}{2}$ False

Step 12

The solution consists of all of the true intervals.

$2 < x \leq \frac{11}{2}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$2 < x \leq \frac{11}{2}$

Interval Notation:

$(2, \frac{11}{2}]$

Step 14

Solve the Inequality for x (3x+1)/(x-2)>=5

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