Solve the Inequality for x (2x-5)/(x-5)-3<0

$\frac{2 x - 5}{x - 5} - 3 < 0$

Step 1

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

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

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

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 3$ by $- 5$ .

$\frac{2 x - 5 - 3 x + 15}{x - 5} < 0$

Subtract $3 x$ from $2 x$ .

$\frac{- x - 5 + 15}{x - 5} < 0$

Add $- 5$ and $15$ .

$\frac{- x + 10}{x - 5} < 0$

Simplify with factoring out.

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

$\frac{- (x) + 10}{x - 5} < 0$

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (x) - 1 (- 10)}{x - 5} < 0$

Factor $- 1$ out of $- (x) - 1 (- 10)$ .

$\frac{- (x - 10)}{x - 5} < 0$

Simplify the expression.

Rewrite $- (x - 10)$ as $- 1 (x - 10)$ .

$\frac{- 1 (x - 10)}{x - 5} < 0$

Move the negative in front of the fraction.

$- \frac{x - 10}{x - 5} < 0$

Step 2

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

$x - 10 = 0$

$x - 5 = 0$

Step 3

Add $10$ to both sides of the equation.

$x = 10$

Step 4

Add $5$ to both sides of the equation.

$x = 5$

Step 5

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

$x = 10$

$x = 5$

Step 6

Consolidate the solutions.

$x = 10, 5$

Step 7

Find the domain of $\frac{2 x - 5}{x - 5} - 3$ .

Set the denominator in $\frac{2 x - 5}{x - 5}$ equal to $0$ to find where the expression is undefined.

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

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

Step 8

Use each root to create test intervals.

$x < 5$

$5 < x < 10$

$x > 10$

Step 9

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

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

$x = 0$

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

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

The left side $- 2$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $5 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $5 < x < 10$ and see if this value makes the original inequality true.

$x = 8$

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

$\frac{2 (8) - 5}{(8) - 5} - 3 < 0$

The left side $0.\overline{6}$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 10$ to see if it makes the inequality true.

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

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

$\frac{2 (12) - 5}{(12) - 5} - 3 < 0$

The left side $- 0.\overline{285714}$ is less than the right side $0$ , which means that the given statement is always true.

True

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

$x < 5$ True

$5 < x < 10$ False

$x > 10$ True

$x < 5$ True

$5 < x < 10$ False

$x > 10$ True

Step 10

The solution consists of all of the true intervals.

$x < 5$ or $x > 10$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < 5 or x > 10$

Interval Notation:

$(- \infty, 5) \cup (10, \infty)$

Step 12

Solve the Inequality for x (2x-5)/(x-5)-3<0

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