Solve the Inequality for x 10/(x-2)>=2

$\frac{10}{x - 2} \geq 2$

Subtract $2$ from both sides of the inequality.

$\frac{10}{x - 2} - 2 \geq 0$

Simplify $\frac{10}{x - 2} - 2$ .

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Combine $- 2$ and $\frac{x - 2}{x - 2}$ .

$\frac{10}{x - 2} + \frac{- 2 (x - 2)}{x - 2} \geq 0$

Combine the numerators over the common denominator.

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

Simplify the numerator.

Factor $2$ out of $10 - 2 (x - 2)$ .

Factor $2$ out of $10$ .

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

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

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

Factor $2$ out of $2 (5) + 2 (- (x - 2))$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

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

Add $5$ and $2$ .

$\frac{2 (- x + 7)}{x - 2} \geq 0$

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

$\frac{2 (- (x) + 7)}{x - 2} \geq 0$

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{2 (- (x) - 1 (- 7))}{x - 2} \geq 0$

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$\frac{2 (- (x - 7))}{x - 2} \geq 0$

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$\frac{2 (- 1 (x - 7))}{x - 2} \geq 0$

Move the negative in front of the fraction.

$- \frac{2 (x - 7)}{x - 2} \geq 0$

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

$x - 7 = 0$

$x - 2 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 7$

$x = 2$

Consolidate the solutions.

$x = 7, 2$

Find the domain of $- \frac{2 (x - 7)}{x - 2}$ .

Set the denominator in $\frac{2 (x - 7)}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$(- \infty, 2) \cup (2, \infty)$

Use each root to create test intervals.

$x < 2$

$2 < x < 7$

$x > 7$

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 $x < 2$ to see if it makes the inequality true.

Choose a value on the interval $x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{10}{(0) - 2} \geq 2$

The left side $- 5$ is less than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $2 < x < 7$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{10}{(4) - 2} \geq 2$

The left side $5$ is greater than the right side $2$ , which means that the given statement is always true.

True

Test a value on the interval $x > 7$ to see if it makes the inequality true.

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

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

$\frac{10}{(10) - 2} \geq 2$

The left side $1.25$ is less than the right side $2$ , which means that the given statement is false.

False

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

$x < 2$ False

$2 < x < 7$ True

$x > 7$ False

$x < 2$ False

$2 < x < 7$ True

$x > 7$ False

The solution consists of all of the true intervals.

$2 < x \leq 7$

The result can be shown in multiple forms.

Inequality Form:

$2 < x \leq 7$

Interval Notation:

$(2, 7]$

Solve the Inequality for x 10/(x-2)>=2

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