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We start solving this just like we would start solving it if it was an equation. We will get one side equal to zero and then factor it.
If this was an equation we would have a product that is zero and we could use the Zero Product Property to solve it.
But we have an inequality. We have a product that is less than 0. In other words, we have a product that is negative. More specifically, we have a product of three factors that is negative.
How do we get a negative answer when we multiply 3 numbers? The product of three numbers is negative if:
All three numbers are negative.
One number is negative and the other two are positive.
Any other combinations will result in a positive or zero product.
So the solutions to our problem will be the x's that make all three factors negative or that make one factor negative and the other two positive. We just have to figure out how to write inequalities that express this.
Writing these inequalities is easier if we first figure out the order of the factors. Choosing from x, (x+3) and (x+7), which is the largest? Which is the smallest? And which is in between? Believe it or not, we can actually answer these questions without knowing what x is! With a little thought I hope it will be clear that (x+7) will always be greater than the other two and x will always be smaller than the other two.
Now that we know the order of the factors it will be simpler to express the inequalities we need:
All three factors are negative. If the largest factor is negative, the other two factors will have to be negative. So if then all three factors will be negative.
One factor is negative and the other two positive. If only one factor is negative then it will have to be the smallest one. And if the middle factor is positive then the largest one will be positive, too. So the inequalities we need are: and . This will make one factor negative and the other two positive.
(If the logic of this is not clear, think about it for a while. Try out some numbers until you can see how this works.) So to say "either all three factors are negative or one factor is negative and the other two are positive" in an algebraic way we need:
x + 7 < 0 or x < 0 and x + 3 > 0
Now we just solve each of the simple inequalities:
x < -7 or (x < 0 and x > -3)
This is our solution. It says that x values that are less than -7 or that are between -3 and 0 are solutions to the original inequality.
Note: The part about determining the order of the factors and using that order to write our inequalities may be hard to see the first few times but it makes the problem much, much easier. Here are the inequalities we would need if we didn't do this:
All three factors are negative:
x < 0 and x +3 < 0 and x + 7 < 0
One factor is positive and two are positive:
(x < 0 and x + 3 > 0 and x + 7 > 0) or (x > 0 and x + 3 < 0 and x + 7 > 0) or (x > 0 and x + 3 > 0 and x + 7 < 0)
So our complete solution would be:
(x < 0 and x +3 < 0 and x + 7 < 0) or ((x < 0 and x + 3 > 0 and x + 7 > 0) or (x > 0 and x + 3 < 0 and x + 7 > 0) or (x > 0 and x + 3 > 0 and x + 7 < 0))
which is much longer and more difficult to solve than what we came up with taking advantage of the order of the factors and some logic:
x + 7 < 0 or (x < 0 and x + 3 > 0)
Similar logic and be used on products that are positive.
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