Solve the Inequality for m 9/(m-5)<3

$\frac{9}{m - 5} < 3$

Subtract $3$ from both sides of the inequality.

$\frac{9}{m - 5} - 3 < 0$

Simplify $\frac{9}{m - 5} - 3$ .

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{m - 5}{m - 5}$ .

$\frac{9}{m - 5} - 3 \cdot \frac{m - 5}{m - 5} < 0$

Combine $- 3$ and $\frac{m - 5}{m - 5}$ .

$\frac{9}{m - 5} + \frac{- 3 (m - 5)}{m - 5} < 0$

Combine the numerators over the common denominator.

$\frac{9 - 3 (m - 5)}{m - 5} < 0$

Simplify the numerator.

Factor $3$ out of $9 - 3 (m - 5)$ .

Factor $3$ out of $9$ .

$\frac{3 (3) - 3 (m - 5)}{m - 5} < 0$

Factor $3$ out of $- 3 (m - 5)$ .

$\frac{3 (3) + 3 (- (m - 5))}{m - 5} < 0$

Factor $3$ out of $3 (3) + 3 (- (m - 5))$ .

$\frac{3 (3 - (m - 5))}{m - 5} < 0$

Apply the distributive property.

$\frac{3 (3 - m - - 5)}{m - 5} < 0$

Multiply $- 1$ by $- 5$ .

$\frac{3 (3 - m + 5)}{m - 5} < 0$

Add $3$ and $5$ .

$\frac{3 (- m + 8)}{m - 5} < 0$

Factor $- 1$ out of $- m$ .

$\frac{3 (- (m) + 8)}{m - 5} < 0$

Rewrite $8$ as $- 1 (- 8)$ .

$\frac{3 (- (m) - 1 (- 8))}{m - 5} < 0$

Factor $- 1$ out of $- (m) - 1 (- 8)$ .

$\frac{3 (- (m - 8))}{m - 5} < 0$

Rewrite $- (m - 8)$ as $- 1 (m - 8)$ .

$\frac{3 (- 1 (m - 8))}{m - 5} < 0$

Move the negative in front of the fraction.

$- \frac{3 (m - 8)}{m - 5} < 0$

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

$m - 8 = 0$

$m - 5 = 0$

Add $8$ to both sides of the equation.

$m = 8$

Add $5$ to both sides of the equation.

$m = 5$

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

$m = 8$

$m = 5$

Consolidate the solutions.

$m = 8, 5$

Find the domain of $- \frac{3 (m - 8)}{m - 5}$ .

Set the denominator in $\frac{3 (m - 8)}{m - 5}$ equal to $0$ to find where the expression is undefined.

$m - 5 = 0$

Add $5$ to both sides of the equation.

$m = 5$

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

$(- \infty, 5) \cup (5, \infty)$

Use each root to create test intervals.

$m < 5$

$5 < m < 8$

$m > 8$

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 $m < 5$ to see if it makes the inequality true.

Choose a value on the interval $m < 5$ and see if this value makes the original inequality true.

$m = 0$

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

$\frac{9}{(0) - 5} < 3$

The left side $- 1.8$ is less than the right side $3$ , which means that the given statement is always true.

True

Test a value on the interval $5 < m < 8$ to see if it makes the inequality true.

Choose a value on the interval $5 < m < 8$ and see if this value makes the original inequality true.

$m = 6$

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

$\frac{9}{(6) - 5} < 3$

The left side $9$ is not less than the right side $3$ , which means that the given statement is false.

False

Test a value on the interval $m > 8$ to see if it makes the inequality true.

Choose a value on the interval $m > 8$ and see if this value makes the original inequality true.

$m = 10$

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

$\frac{9}{(10) - 5} < 3$

The left side $1.8$ is less than the right side $3$ , which means that the given statement is always true.

True

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

$m < 5$ True

$5 < m < 8$ False

$m > 8$ True

$m < 5$ True

$5 < m < 8$ False

$m > 8$ True

The solution consists of all of the true intervals.

$m < 5$ or $m > 8$

The result can be shown in multiple forms.

Inequality Form:

$m < 5 or m > 8$

Interval Notation:

$(- \infty, 5) \cup (8, \infty)$

Solve the Inequality for m 9/(m-5)<3

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