Solve the Inequality for x 1/(3x-2)<=4

$\frac{1}{3 x - 2} \leq 4$

Step 1

Subtract $4$ from both sides of the inequality.

$\frac{1}{3 x - 2} - 4 \leq 0$

Step 2

Simplify $\frac{1}{3 x - 2} - 4$ .

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3 x - 2}{3 x - 2}$ .

$\frac{1}{3 x - 2} - 4 \cdot \frac{3 x - 2}{3 x - 2} \leq 0$

Combine $- 4$ and $\frac{3 x - 2}{3 x - 2}$ .

$\frac{1}{3 x - 2} + \frac{- 4 (3 x - 2)}{3 x - 2} \leq 0$

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{1 - 4 (3 x) - 4 \cdot - 2}{3 x - 2} \leq 0$

Multiply $3$ by $- 4$ .

$\frac{1 - 12 x - 4 \cdot - 2}{3 x - 2} \leq 0$

Multiply $- 4$ by $- 2$ .

$\frac{1 - 12 x + 8}{3 x - 2} \leq 0$

Add $1$ and $8$ .

$\frac{- 12 x + 9}{3 x - 2} \leq 0$

Factor $3$ out of $- 12 x + 9$ .

Factor $3$ out of $- 12 x$ .

$\frac{3 (- 4 x) + 9}{3 x - 2} \leq 0$

Factor $3$ out of $9$ .

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

Factor $3$ out of $3 (- 4 x) + 3 (3)$ .

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

Factor $- 1$ out of $- 4 x$ .

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

Rewrite $3$ as $- 1 (- 3)$ .

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

Factor $- 1$ out of $- (4 x) - 1 (- 3)$ .

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

Rewrite $- (4 x - 3)$ as $- 1 (4 x - 3)$ .

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

Move the negative in front of the fraction.

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

Step 3

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

$4 x - 3 = 0$

$3 x - 2 = 0$

Step 4

Add $3$ to both sides of the equation.

$4 x = 3$

Step 5

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 6

Add $2$ to both sides of the equation.

$3 x = 2$

Step 7

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 8

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

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

$x = \frac{2}{3}$

Step 9

Consolidate the solutions.

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

Step 10

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

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

$3 x - 2 = 0$

Solve for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

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

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

Step 11

Use each root to create test intervals.

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

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

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

Step 12

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 $x < \frac{2}{3}$ to see if it makes the inequality true.

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

$x = 0$

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

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

The left side $- 0.5$ is less than the right side $4$ , which means that the given statement is always true.

True

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

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

$x = 0.71$

Replace $x$ with $0.71$ in the original inequality.

$\frac{1}{3 (0.71) - 2} \leq 4$

The left side $7.\overline{692307}$ is greater than the right side $4$ , which means that the given statement is false.

False

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

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

$x = 3$

Replace $x$ with $3$ in the original inequality.

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

The left side $0.\overline{142857}$ is less than the right side $4$ , which means that the given statement is always true.

True

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

$x < \frac{2}{3}$ True

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

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

$x < \frac{2}{3}$ True

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

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

Step 13

The solution consists of all of the true intervals.

$x < \frac{2}{3}$ or $x \geq \frac{3}{4}$

Step 14

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{2}{3} or x \geq \frac{3}{4}$

Interval Notation:

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

Step 15

Solve the Inequality for x 1/(3x-2)<=4

Related Questions

Support LCMGCF.com by disabling your adblocker.