Convert to Set Notation ((x-4)(x+5))/(x-6)<=0

$\frac{(x - 4) (x + 5)}{x - 6} \leq 0$

Step 1

Solve $x \leq - 5 or 4 \leq x < 6$ .

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

$x - 4 = 0$

$x + 5 = 0$

$x - 6 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Subtract $5$ from both sides of the equation.

$x = - 5$

Add $6$ to both sides of the equation.

$x = 6$

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

$x = 4$

$x = - 5$

$x = 6$

Consolidate the solutions.

$x = 4, - 5, 6$

Find the domain of $\frac{(x - 4) (x + 5)}{x - 6}$ .

Set the denominator in $\frac{(x - 4) (x + 5)}{x - 6}$ equal to $0$ to find where the expression is undefined.

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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

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

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 4$

$4 < x < 6$

$x > 6$

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 $x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

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

$\frac{((- 8) - 4) ((- 8) + 5)}{(- 8) - 6} \leq 0$

The left side $- 2.\overline{571428}$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 5 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{((0) - 4) ((0) + 5)}{(0) - 6} \leq 0$

The left side $3.\overline{3}$ is greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $4 < x < 6$ to see if it makes the inequality true.

Choose a value on the interval $4 < x < 6$ and see if this value makes the original inequality true.

$x = 5$

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

$\frac{((5) - 4) ((5) + 5)}{(5) - 6} \leq 0$

The left side $- 10$ is less than the right side $0$ , 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.

$\frac{((8) - 4) ((8) + 5)}{(8) - 6} \leq 0$

The left side $26$ 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.

$x < - 5$ True

$- 5 < x < 4$ False

$4 < x < 6$ True

$x > 6$ False

$x < - 5$ True

$- 5 < x < 4$ False

$4 < x < 6$ True

$x > 6$ False

The solution consists of all of the true intervals.

$x \leq - 5$ or $4 \leq x < 6$

Step 2

Use the inequality $x \leq - 5 or 4 \leq x < 6$ to build the set notation.

${x | x \leq - 5 or 4 \leq x < 6}$

Step 3

Convert to Set Notation ((x-4)(x+5))/(x-6)<=0

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