What is the value of a if (a-7) (a-11) >0?

Answer 1

For this expression to be >0 it has to be positive so either (a-7) and (a-11) are both positive which occurs when a>11 or both terms are negative which occurs when a< 7. So our solution is 7>a>11.

You can prove this to yourself by substituting a number >11 or less than 7 and you will see that the product is a positive number. Then try a number in between such as 9 and you will get a negative number

Answer 2

If it were an equation, the two solutions are a=7 and a=11.

That is, the function is zero when a=7 and a=11.

So there are three parts of the number line than need investigated:

a<7 , 7 < a < 11, and a>11

or

(-infinity, 7) (7,11) and (11, infinity)

For the first part, (infinity, 7) we choose a sample test point for a, like a=0.

(0-7)(0-11) = (-7)(-11) = 77 which is positive, so this part of the number line

is in the solution.

For the second part (7,11) we choose another sample test point for a, like a=10.

(10-7)(10-11) = (3)(-1) = -3 < 0, so this part of the number line is NOT included

in the solution.

Finally, for the third section, we choose again another sample point like a=15.

(15-7)(15-11) = 8*7 = 56 > 0, which is positive, so the third section is part of

the solution.

The solution set is therefore (-infinity, 7) U (11,infinity)