Convert to Interval Notation (x^2-4x)/(x+5)>0

Problem

(x2−4*x)/(x+5)>0

Solution

1. Factor the numerator to identify the zeros of the expression.

x2−4*x=x*(x−4)

2. Identify the critical points where the expression is zero or undefined by setting each factor in the numerator and denominator to zero.

x=0

x−4=0⇒x=4

x+5=0⇒x=−5

3. Determine the intervals created by these critical points on the number line.

(−∞,−5),(−5,0),(0,4),(4,∞)

4. Test a value from each interval in the inequality (x*(x−4))/(x+5)>0 to check the sign.
   For (−∞,−5) test x=−6 (−6*(−10))/(−1)=−60 (Negative)
   For (−5,0) test x=−1 (−1*(−5))/4=5/4 (Positive)
   For (0,4) test x=1 (1*(−3))/6=−1/2 (Negative)
   For (4,∞) test x=5 (5*(1))/10=1/2 (Positive)

5. Select the intervals where the expression is greater than zero.

(−5,0)∪(4,∞)

Final Answer

(x2−4*x)/(x+5)>0⇒(−5,0)∪(4,∞)

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