Your inequality says that the product (multiplication) of three numbers is less than zero. In other words the product is negative. If you understand multiplication, you will realize that there are only two ways to get a negative answer when you multiply three numbers:- All three numbers are negative
- Two numbers are positive and one is negative
There are solutions from both possibilities. We will solve each possibility in order
1. All three numbers are negative.
The easiest way to find this solution is to figure out which of the three numbers is going to be the largest. Even though we do not know what number "b" will be we can still figure out which factor is the largest. If you think about (b+3), (b-5) and (b-6), it should be clear that, no matter what number "b" is, (b+3) will end up being larger than (b-3) or (b-6).
Now think of this: If (b+3) is the largest of the three and it is negative, doesn't that mean that the two smaller numbers also have to be negative? Think of a number line. On a number line smaller numbers are to the left. If (b+3) is negative and if (b-3) and (b-6) are smaller numbers, won't (b-3) and (b-6) also be to the left of both (b+3) and 0?
Now we know that if (b+3) is negative, all three must be negative. So the solution for "All three numbers are negative" will be the solution to:
b+3 < 0
Subtracting 3 from both sides:
b < -3
This is part one of the overall solution.
2. Two numbers are positive and one is negative. Using logic similar to the logic used to determine the largest factor we can determine that (b-6) must be the smallest factor and (b-3) must be the "in between" factor.
Now how do we get two positives and one negative? Well the negative factor must be the smallest factor, (b-6). The other two must be positive. Using logic like above we can figure out that if (b-3) is positive then (b+3) must also be positive. So to get two positive factors and one negative factor:
b-6 < 0 and b-3 > 0
Adding 6 to both sides of the first inequality and adding 3 to both sides of the second we get:
b < 6 and b > 3
This says that "b" must be between 3 and 6 (not including 3 or 6). This is the second part of our solution.
The complete solution is that "b" must be less than -3 (part 1) or "b" must be between 3 and 6 (part 2).