algebra 2 problem

Question

How do I solve this inequality x^2(9+x)(x-9)/(x+4)(x-7) ≥0

Answer 1

The best way to do this is to find the zeros and restrictions, since we have a denominator.  We will then use those values as references for test points to see which values of x make the expression equal to zero or positive.

 

Here are our zeroes.

 

x²(9 + x)(x - 9) = 0

 

x = 0 , x = -9 , and x = 9

 

 

 

Here are our restrictions.

 

(x + 4)(x - 7) = 0

 

x = -4  ,  x = 7

 

 

Therefore, the reference values we will use for test points are

 

x=-9 , x=-4 , x=0 , x=7 , x=9

 

 

Evaluate the expression when x=-10 , x=-6 , x=-1 , x=1 , x=8 , x=10   in this specific order.

 

 

After you evaluate the expression, pick the intervals of x where the evaluated result is equal to zero or positive.  I will do the first two for you.

 

(-10)²(9 - 10)(-10 - 9) / (-10 + 4)(-10 - 7) = (100)(-1)(-19) / (-6)(-17) = positive number

 

(-6)²(9 - 6)(-6 - 9) / (-6 + 4)(-6 - 7) = (36)(3)(-15) / (-2)(-13) = negative number

 

 

Complete the last 4 evaluations to get your intervals.