Question 609184
In essence your inequality says that the product of three numbers (x+10, x-2 and x+7) is greater than zero. In other words, the product of three numbers is positive. Our solution will be based on understanding how the product of three numbers can turn out positive. With a little thought I hope you can see that it can only happen if a) all three numbers are positive; or b) two numbers of negative and the third one is positive.<br>
For each of these possibilities there is a straightforward but more difficult solution and a clever, easier solution. The clever solutions are based on figuring out which factor is largest, which one is smallest and which one is in between the other two. With a little thought I hope you can see that, no matter what number "x" turns out to be:
x+10 must be the largest factor, and
x-2 must be the smallest factor, and
x+7 must be the in between factor.<br>
Now let's look at writing expressions to solve for...
a) All three factors are positive
The straightforward method would be to say
x+10 > 0 and x-2 > 0 and x+7 > 0
The clever way would be to realize that if the smallest factor is positive the other two factors would have to be positive, too! So we can use just
x-2 > 0
instead of
x+10 > 0 and x-2 > 0 and x+7 > 0
Solving the clever one we get:
x > 2<br>
b) Two negative factors and one positive one. The straightforward way would be to write inequalities that express all the possible combinations of two negative and one positive factor:
(x+10 < 0 and x-2 < 0 and x+7 > 0) or (x+10 < 0 and x-2 > 0 and x+7 < 0) or (x+10 > 0 and x-2 < 0 and x+7 < 0)
The clever way is to realize that the positive factor has to be the the largest one and that the other factors will both be negative if the in between one is negative. So we can use:
x+10 > 0 and x+7 < 0
instead of the big mess above. Solving the clever one we get:
x > -10 and x < -7<br>
Since either a) or b) give us solutions, the full solution to your problem is:
x > 2 or (x > -10 and x < -7)<br>
NOTE 1: The straightforward solutions work out <i>exactly</i> the same as the clever ones. They are just more difficult and time-consuming.
NOTE 2: If you don't really understand the clever solutions, then you'll have to resign yourself to the straightforward solutions.