Question

For what values of x is the product \[\left( {x + 4} \right)\left( {x + 6} \right)\] positive?

Answer 1

Hint: We are given two brackets. In both the brackets we have a variable x. Now in order to find that the product should be positive, we need to have both the brackets either positive or negative. So we will equate it to greater than zero such that the product is positive.

Complete step by step answer:
Given that, \[\left( {x + 4} \right)\left( {x + 6} \right)\] is positive.
Thus \[\left( {x + 4} \right)\left( {x + 6} \right) > 0\]
Now we will separately equate the brackets.
\[\left( {x + 4} \right) > 0\]
So when we transpose the constant numbers we get, \[x > - 4\]
Now we will check for \[\left( {x + 6} \right) > 0\]
On transposing the constant number we get, \[x > - 6\]
Thus we can clearly say that the values of x should be greater than -4
Or
If we take both the terms negative then,
\[\left( {x + 4} \right) < 0\] and \[\left( {x + 6} \right) < 0\]
So when we transpose the constant numbers we get,
\[x < - 4\] and \[x < - 6\]
From here we can say that x should be less than -6.

Note:
This question is very simple to answer actually. We can also use the quadratic equation or quadratic formula method. In which we can find the values of x only by multiplying these brackets and finding the values using a quadratic formula. But that will not give the range or more than two values of x that can be used above.
Also note that we can simply use x=0 also because that will give two positive numbers which on multiplying gives a positive number.