Convert to Interval Notation x-24/x<2

$x - \frac{24}{x} < 2$

Step 1

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $x - \frac{24}{x} < 2$ by $x$ to eliminate the fractions.

Multiply each term in $x - \frac{24}{x} < 2$ by $x$ .

$x \cdot x - \frac{24}{x} x < 2 x$

Simplify the left side.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - \frac{24}{x} x < 2 x$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{24}{x}$ into the numerator.

$x^{2} + \frac{- 24}{x} x < 2 x$

Cancel the common factor.

$x^{2} + \frac{- 24}{x} x < 2 x$

Rewrite the expression.

$x^{2} - 24 < 2 x$

Step 3

Solve the inequality.

Subtract $2 x$ from both sides of the inequality.

$x^{2} - 24 - 2 x < 0$

Convert the inequality to an equation.

$x^{2} - 24 - 2 x = 0$

Factor $x^{2} - 24 - 2 x$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$(x - 6) (x + 4) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 4 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

Step 4

Find the domain of $x - \frac{24}{x} - 2$ .

Set the denominator in $\frac{24}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

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

$(- \infty, 0) \cup (0, \infty)$

Step 5

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 0$

$0 < x < 6$

$x > 6$

Step 6

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 $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Replace $x$ with $- 6$ in the original inequality.

$(- 6) - \frac{24}{- 6} < 2$

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

True

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

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

$(- 2) - \frac{24}{- 2} < 2$

The left side $10$ is not less than the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < 6$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$(3) - \frac{24}{3} < 2$

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

True

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.

$(8) - \frac{24}{8} < 2$

The left side $5$ is not 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.

$x < - 4$ True

$- 4 < x < 0$ False

$0 < x < 6$ True

$x > 6$ False

$x < - 4$ True

$- 4 < x < 0$ False

$0 < x < 6$ True

$x > 6$ False

Step 7

The solution consists of all of the true intervals.

$x < - 4$ or $0 < x < 6$

Step 8

Convert the inequality to interval notation.

$(- \infty, - 4) \cup (0, 6)$

Step 9

Convert to Interval Notation x-24/x<2

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