Solve for x (x-5)/(-x-1)<0

Problem

(x−5)/(−x−1)<0

Solution

1. Identify the critical values by finding where the numerator and denominator are equal to zero.

Numerator: *x−5=0⇒x=5

Denominator: −x−1=0⇒x=−1

2. Determine the intervals created by these critical values on the number line.

Intervals: *(−∞,−1),(−1,5),(5,∞)

3. Test a value from each interval in the inequality (x−5)/(−x−1)<0 to check the sign.

For *(−∞,−1), test *x=−2:(−2−5)/(−(−2)−1)=(−7)/1=−7<0* (True)

For *(−1,5), test *x=0:(0−5)/(−0−1)=(−5)/(−1)=5<0* (False)

For *(5,∞), test *x=6:(6−5)/(−6−1)=1/(−7)=−1/7<0* (True)

4. Combine the intervals where the inequality holds true.

Solution: *x<−1* or *x>5

Final Answer

(x−5)/(−x−1)<0⇒x∈(−∞,−1)∪(5,∞)

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