Solve the Inequality for x (3x+2)/(x-4)<=7

$\frac{3 x + 2}{x - 4} \leq 7$

Subtract $7$ from both sides of the inequality.

$\frac{3 x + 2}{x - 4} - 7 \leq 0$

Simplify $\frac{3 x + 2}{x - 4} - 7$ .

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{3 x + 2}{x - 4} - 7 \cdot \frac{x - 4}{x - 4} \leq 0$

Combine $- 7$ and $\frac{x - 4}{x - 4}$ .

$\frac{3 x + 2}{x - 4} + \frac{- 7 (x - 4)}{x - 4} \leq 0$

Combine the numerators over the common denominator.

$\frac{3 x + 2 - 7 (x - 4)}{x - 4} \leq 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 2 - 7 x - 7 \cdot - 4}{x - 4} \leq 0$

Multiply $- 7$ by $- 4$ .

$\frac{3 x + 2 - 7 x + 28}{x - 4} \leq 0$

Subtract $7 x$ from $3 x$ .

$\frac{- 4 x + 2 + 28}{x - 4} \leq 0$

Add $2$ and $28$ .

$\frac{- 4 x + 30}{x - 4} \leq 0$

Factor $2$ out of $- 4 x + 30$ .

Factor $2$ out of $- 4 x$ .

$\frac{2 (- 2 x) + 30}{x - 4} \leq 0$

Factor $2$ out of $30$ .

$\frac{2 (- 2 x) + 2 (15)}{x - 4} \leq 0$

Factor $2$ out of $2 (- 2 x) + 2 (15)$ .

$\frac{2 (- 2 x + 15)}{x - 4} \leq 0$

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

$\frac{2 (- (2 x) + 15)}{x - 4} \leq 0$

Rewrite $15$ as $- 1 (- 15)$ .

$\frac{2 (- (2 x) - 1 (- 15))}{x - 4} \leq 0$

Factor $- 1$ out of $- (2 x) - 1 (- 15)$ .

$\frac{2 (- (2 x - 15))}{x - 4} \leq 0$

Rewrite $- (2 x - 15)$ as $- 1 (2 x - 15)$ .

$\frac{2 (- 1 (2 x - 15))}{x - 4} \leq 0$

Move the negative in front of the fraction.

$- \frac{2 (2 x - 15)}{x - 4} \leq 0$

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

$2 x - 15 = 0$

$x - 4 = 0$

Add $15$ to both sides of the equation.

$2 x = 15$

Divide each term in $2 x = 15$ by $2$ and simplify.

Divide each term in $2 x = 15$ by $2$ .

$\frac{2 x}{2} = \frac{15}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{15}{2}$

Divide $x$ by $1$ .

$x = \frac{15}{2}$

Add $4$ to both sides of the equation.

$x = 4$

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

$x = \frac{15}{2}$

$x = 4$

Consolidate the solutions.

$x = \frac{15}{2}, 4$

Find the domain of $- \frac{2 (2 x - 15)}{x - 4}$ .

Set the denominator in $\frac{2 (2 x - 15)}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

Use each root to create test intervals.

$x < 4$

$4 < x < \frac{15}{2}$

$x > \frac{15}{2}$

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.

$x = 0$

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

$\frac{3 (0) + 2}{(0) - 4} \leq 7$

The left side $- 0.5$ is less than the right side $7$ , which means that the given statement is always true.

True

Test a value on the interval $4 < x < \frac{15}{2}$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < \frac{15}{2}$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$\frac{3 (6) + 2}{(6) - 4} \leq 7$

The left side $10$ is greater than the right side $7$ , which means that the given statement is false.

False

Test a value on the interval $x > \frac{15}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{15}{2}$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$\frac{3 (10) + 2}{(10) - 4} \leq 7$

The left side $5.\overline{3}$ is less than the right side $7$ , which means that the given statement is always true.

True

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

$x < 4$ True

$4 < x < \frac{15}{2}$ False

$x > \frac{15}{2}$ True

$x < 4$ True

$4 < x < \frac{15}{2}$ False

$x > \frac{15}{2}$ True

The solution consists of all of the true intervals.

$x < 4$ or $x \geq \frac{15}{2}$

The result can be shown in multiple forms.

Inequality Form:

$x < 4 or x \geq \frac{15}{2}$

Interval Notation:

$(- \infty, 4) \cup [\frac{15}{2}, \infty)$

Solve the Inequality for x (3x+2)/(x-4)<=7

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