Solve the Inequality for x 6/(x-1)>=5

$\frac{6}{x - 1} \geq 5$

Step 1

Subtract $5$ from both sides of the inequality.

$\frac{6}{x - 1} - 5 \geq 0$

Step 2

Simplify $\frac{6}{x - 1} - 5$ .

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{6}{x - 1} - 5 \cdot \frac{x - 1}{x - 1} \geq 0$

Combine $- 5$ and $\frac{x - 1}{x - 1}$ .

$\frac{6}{x - 1} + \frac{- 5 (x - 1)}{x - 1} \geq 0$

Combine the numerators over the common denominator.

$\frac{6 - 5 (x - 1)}{x - 1} \geq 0$

Simplify the numerator.

Apply the distributive property.

$\frac{6 - 5 x - 5 \cdot - 1}{x - 1} \geq 0$

Multiply $- 5$ by $- 1$ .

$\frac{6 - 5 x + 5}{x - 1} \geq 0$

Add $6$ and $5$ .

$\frac{- 5 x + 11}{x - 1} \geq 0$

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 11}{x - 1} \geq 0$

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

$\frac{- (5 x) - 1 (- 11)}{x - 1} \geq 0$

Factor $- 1$ out of $- (5 x) - 1 (- 11)$ .

$\frac{- (5 x - 11)}{x - 1} \geq 0$

Rewrite $- (5 x - 11)$ as $- 1 (5 x - 11)$ .

$\frac{- 1 (5 x - 11)}{x - 1} \geq 0$

Move the negative in front of the fraction.

$- \frac{5 x - 11}{x - 1} \geq 0$

Step 3

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

$5 x - 11 = 0$

$x - 1 = 0$

Step 4

Add $11$ to both sides of the equation.

$5 x = 11$

Step 5

Divide each term in $5 x = 11$ by $5$ and simplify.

Divide each term in $5 x = 11$ by $5$ .

$\frac{5 x}{5} = \frac{11}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{11}{5}$

Divide $x$ by $1$ .

$x = \frac{11}{5}$

Step 6

Add $1$ to both sides of the equation.

$x = 1$

Step 7

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

$x = \frac{11}{5}$

$x = 1$

Step 8

Consolidate the solutions.

$x = \frac{11}{5}, 1$

Step 9

Find the domain of $- \frac{5 x - 11}{x - 1}$ .

Set the denominator in $\frac{5 x - 11}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$(- \infty, 1) \cup (1, \infty)$

Step 10

Use each root to create test intervals.

$x < 1$

$1 < x < \frac{11}{5}$

$x > \frac{11}{5}$

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 $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{6}{(0) - 1} \geq 5$

The left side $- 6$ is less than the right side $5$ , which means that the given statement is false.

False

Test a value on the interval $1 < x < \frac{11}{5}$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < \frac{11}{5}$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$\frac{6}{(2) - 1} \geq 5$

The left side $6$ is greater than the right side $5$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{11}{5}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{11}{5}$ and see if this value makes the original inequality true.

$x = 5$

Replace $x$ with $5$ in the original inequality.

$\frac{6}{(5) - 1} \geq 5$

The left side $1.5$ 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.

$x < 1$ False

$1 < x < \frac{11}{5}$ True

$x > \frac{11}{5}$ False

$x < 1$ False

$1 < x < \frac{11}{5}$ True

$x > \frac{11}{5}$ False

Step 12

The solution consists of all of the true intervals.

$1 < x \leq \frac{11}{5}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$1 < x \leq \frac{11}{5}$

Interval Notation:

$(1, \frac{11}{5}]$

Step 14

Solve the Inequality for x 6/(x-1)>=5

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