Convert to Set Notation ((x+12)(x-14))/(x+10)<=0

$\frac{(x + 12) (x - 14)}{x + 10} \leq 0$

Step 1

Solve $x \leq - 12 or - 10 < x \leq 14$ .

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

$x + 12 = 0$

$x - 14 = 0$

$x + 10 = 0$

Subtract $12$ from both sides of the equation.

$x = - 12$

Add $14$ to both sides of the equation.

$x = 14$

Subtract $10$ from both sides of the equation.

$x = - 10$

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

$x = - 12$

$x = 14$

$x = - 10$

Consolidate the solutions.

$x = - 12, 14, - 10$

Find the domain of $\frac{(x + 12) (x - 14)}{x + 10}$ .

Set the denominator in $\frac{(x + 12) (x - 14)}{x + 10}$ equal to $0$ to find where the expression is undefined.

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

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

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

Use each root to create test intervals.

$x < - 12$

$- 12 < x < - 10$

$- 10 < x < 14$

$x > 14$

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 $x < - 12$ to see if it makes the inequality true.

Choose a value on the interval $x < - 12$ and see if this value makes the original inequality true.

$x = - 14$

Replace $x$ with $- 14$ in the original inequality.

$\frac{((- 14) + 12) ((- 14) - 14)}{(- 14) + 10} \leq 0$

The left side $- 14$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 12 < x < - 10$ to see if it makes the inequality true.

Choose a value on the interval $- 12 < x < - 10$ and see if this value makes the original inequality true.

$x = - 11$

Replace $x$ with $- 11$ in the original inequality.

$\frac{((- 11) + 12) ((- 11) - 14)}{(- 11) + 10} \leq 0$

The left side $25$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 10 < x < 14$ to see if it makes the inequality true.

Choose a value on the interval $- 10 < x < 14$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{((0) + 12) ((0) - 14)}{(0) + 10} \leq 0$

The left side $- 16.8$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 14$ to see if it makes the inequality true.

Choose a value on the interval $x > 14$ and see if this value makes the original inequality true.

$x = 16$

Replace $x$ with $16$ in the original inequality.

$\frac{((16) + 12) ((16) - 14)}{(16) + 10} \leq 0$

The left side $2.\overline{153846}$ is greater than the right side $0$ , which means that the given statement is false.

False

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

$x < - 12$ True

$- 12 < x < - 10$ False

$- 10 < x < 14$ True

$x > 14$ False

$x < - 12$ True

$- 12 < x < - 10$ False

$- 10 < x < 14$ True

$x > 14$ False

The solution consists of all of the true intervals.

$x \leq - 12$ or $- 10 < x \leq 14$

Step 2

Use the inequality $x \leq - 12 or - 10 < x \leq 14$ to build the set notation.

${x | x \leq - 12 or - 10 < x \leq 14}$

Step 3

Convert to Set Notation ((x+12)(x-14))/(x+10)<=0

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