This says that the product of 3 numbers (x, x-6 and x+2) is greater than zero. In other words the product of three numbers is positive. The key to the solution is the answer to the question: "How can is it possible to get a positive result when you multiply 3 numbers?" I hope the answer is obvious to you. You get a positive result when multiplying 3 numbers when a) all 3 numbers are positive, or b) two of the numbers are negative and the third one is positive. Any other combination of 3 numbers will have a negative or zero result. So the solution to this inequality will be all the x values that make
all 3 factors positive, or
make two of the factors negative and the third one positive
Now we just have to figure out how to translate this idea into number sentences. Making this translation is easier if we take time to figure out the order of the factors, if possible. Which one is the largest? Which one is the smallest? And which one is the middle factor? Even though we do not know what value x may be, it is possible, with these 3 factors, to know the order of them. After thinking about this for a while I hope you can see that (x+2) will always be greater than the other two factors, no matter what x happens to be. Similarly (x-6) will always be less than the other two no matter what x happens to be. And that makes x the in between factor. If you have trouble seeing this:
Picture a number line.
Picture any random point on this number line. (This is our "x").
x+2 will be two units to the right of x. Being to the right makes x+2 greater than x.
x-6 will be 6 units to the left of x. Being to the left makes x-6 less than x.
With x-6 to the left and x+2 on the right, x is the in-between number.
Knowing the order of the factors, combined with some logic, will make expressing the ideas "the three factors are positive" and "two factors are negative and one is positive" easier.
"The three factors are positive"
Without using the order or any logic we could say:
x+%3E+0 and x-6+%3E+0 and x+%2B+2+%3E+0
This is not terribly difficult to solve. But if we use the order and some logic we can come up with something much easier. Think about this: If the lowest of three numbers is positive, wouldn't the other two have to be positive, too? In other words, if the lowest of three numbers is to the right of zero on the number line, wouldn't the other two numbers, which are to the right of the lowest, also be to the right of zero? I hope you can see the logic of this. So to make all three factors positive we just have to make sure the lowest one is positive:
x-6+%3E+0
"Two factors are negative and one is positive"
Without using the order or any logic we would have to describe every possible combination of two negative factors and one positive factor:
((x+%3C+0) and x+-+6+%3C+0 and x+%2B+2+%3E+0) or (x+%3C+0) and x+-+6+%3E+0 and x+%2B+2+%3C+0) or (x+%3E+0) and x+-+6+%3C+0 and x+%2B+2+%3C+0))
Yikes!! This is solvable but it will take a lot of work. But if we use the order sand some logic we can get something much simpler to solve. When we know the order of the factors, in order to have 2 of three factors negative and one of them positive we only have to make sure that the in between factor is negative and the largest factor is positive. This is so because if the in-between factor is negative then the smallest of the three would have to be negative, too. So to make two factors negative and one positive all we have to say is:
(x+%3C+0 and x+%2B+2+%3E+0)
I hope you agree that using the order and some logic makes this problem much easier to solve. When you know the order of the factors this way is much easier. (Note: if it is not possible to figure out the order of the factors, then you must use the much longer, more complicated compound inequalities.)
Remember that our solution is the set of x values that make "all three factors positive or make two factors negative and one positive". Using what we figured out above this is:
(x+-6+%3E+0) or (x+%3C+0 and x+%2B+2+%3E+0)
Now we solve this compound inequality:
(x+%3E+6) or (x+%3C+0 and x+%3E+-2)
In interval notation (x+%3E+6) is (6, infinity)
In interval notation (x+%3C+0 and x+%3E+-2) is (-2, 0)
As for "My teacher is also asking me to find the vertex, axis and symmetry.": I agree, WHAT??. Even though I think it was phrased "axis of symmetry" not "axis and symmetry" it still doesn't make sense. The words "vertex", "axis of symmetry" apply to graphs of two variable sentences (usually parabolas). But we do not have a y. And if the inequality was x%28x-6%29%28x%2B2%29+%3E+y it would simplify to x%5E3+-4x%5E2+-12x+%3C+y. This is a cubic inequality not a quadratic inequality. Its graph would not be a parabola, would not have a vertex and would not have an axis of symmetry.
Step-by-step explanation:
