Solve the Inequality for x x-10/(x-1)>=4

$x - \frac{10}{x - 1} \geq 4$

Step 1

Subtract $4$ from both sides of the inequality.

$x - \frac{10}{x - 1} - 4 \geq 0$

Step 2

Simplify $x - \frac{10}{x - 1} - 4$ .

Find the common denominator.

Write $x$ as a fraction with denominator $1$ .

$\frac{x}{1} - \frac{10}{x - 1} - 4 \geq 0$

Multiply $\frac{x}{1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x}{1} \cdot \frac{x - 1}{x - 1} - \frac{10}{x - 1} - 4 \geq 0$

Multiply $\frac{x}{1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x (x - 1)}{x - 1} - \frac{10}{x - 1} - 4 \geq 0$

Write $- 4$ as a fraction with denominator $1$ .

$\frac{x (x - 1)}{x - 1} - \frac{10}{x - 1} + \frac{- 4}{1} \geq 0$

Multiply $\frac{- 4}{1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x (x - 1)}{x - 1} - \frac{10}{x - 1} + \frac{- 4}{1} \cdot \frac{x - 1}{x - 1} \geq 0$

Multiply $\frac{- 4}{1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x (x - 1)}{x - 1} - \frac{10}{x - 1} + \frac{- 4 (x - 1)}{x - 1} \geq 0$

Combine the numerators over the common denominator.

$\frac{x (x - 1) - 10 - 4 (x - 1)}{x - 1} \geq 0$

Simplify each term.

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 - 10 - 4 (x - 1)}{x - 1} \geq 0$

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 1 - 10 - 4 (x - 1)}{x - 1} \geq 0$

Move $- 1$ to the left of $x$ .

$\frac{x^{2} - 1 \cdot x - 10 - 4 (x - 1)}{x - 1} \geq 0$

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} - x - 10 - 4 (x - 1)}{x - 1} \geq 0$

Apply the distributive property.

$\frac{x^{2} - x - 10 - 4 x - 4 \cdot - 1}{x - 1} \geq 0$

Multiply $- 4$ by $- 1$ .

$\frac{x^{2} - x - 10 - 4 x + 4}{x - 1} \geq 0$

Subtract $4 x$ from $- x$ .

$\frac{x^{2} - 5 x - 10 + 4}{x - 1} \geq 0$

Add $- 10$ and $4$ .

$\frac{x^{2} - 5 x - 6}{x - 1} \geq 0$

Factor $x^{2} - 5 x - 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

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

Step 3

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

$x - 6 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 4

Add $6$ to both sides of the equation.

$x = 6$

Step 5

Subtract $1$ from both sides of the equation.

$x = - 1$

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.

$x = 6$

$x = - 1$

$x = 1$

Step 8

Consolidate the solutions.

$x = 6, - 1, 1$

Step 9

Find the domain of $\frac{(x - 6) (x + 1)}{x - 1}$ .

Set the denominator in $\frac{(x - 6) (x + 1)}{x - 1}$ 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.

$x < - 1$

$- 1 < x < 1$

$1 < x < 6$

$x > 6$

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.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$(- 4) - \frac{10}{(- 4) - 1} \geq 4$

The left side $- 2$ is less than the right side $4$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 1$ to see if it makes the inequality true.

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

$x = 0$

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

$(0) - \frac{10}{(0) - 1} \geq 4$

The left side $10$ is greater than the right side $4$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 6$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$(4) - \frac{10}{(4) - 1} \geq 4$

The left side $0.\overline{6}$ is less than the right side $4$ , which means that the given statement is false.

False

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

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

$(8) - \frac{10}{(8) - 1} \geq 4$

The left side $6.\overline{571428}$ is greater 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.

$x < - 1$ False

$- 1 < x < 1$ True

$1 < x < 6$ False

$x > 6$ True

$x < - 1$ False

$- 1 < x < 1$ True

$1 < x < 6$ False

$x > 6$ True

Step 12

The solution consists of all of the true intervals.

$- 1 \leq x < 1$ or $x \geq 6$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x < 1 or x \geq 6$

Interval Notation:

$[- 1, 1) \cup [6, \infty)$

Step 14

Solve the Inequality for x x-10/(x-1)>=4

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