What part of it
do you understand and what part are you asking about? For example, do you need an explanation for why "3= 1+ 2"? (I'm hoping not!
)
Do you know that, to add fractions, you need to get a "common denominator"?
The first step, from \(\displaystyle x^2+ 2x+ 3\) to \(\displaystyle (x+ 1)^2+ 2\) is "completing the square". You should know that \(\displaystyle (x+ a)^2= x^2+ 2ax+ a^2\). Comparing the first two terms, \(\displaystyle x^2+ 2x\) to \(\displaystyle x^2+ 2ax\) you should see that 2x= 2ax so a= 1. In that case, \(\displaystyle (x+ 1)^2= x^2+ 2a+ 1\) so that you can write \(\displaystyle x^2+ 2x+ 3= x^2+ 2x+ 1+ 2= (x+ 1)^2+ 2\). (See? We needed to know that 3= 1+ 2!) Since a square is never negative and we are adding 2 to a "perfect square", \(\displaystyle x^2+ 2x+ 3\) is never negative (in fact it is never less than 2).
The next part is "adding fractions by getting a common denominator". The fraction \(\displaystyle \frac{x}{2x+ 3}\) has denominator "2x+ 3". The fraction \(\displaystyle \frac{1}{x}\) has denominator x. We change the denominators of fractions by
multiplying both numerator and denominator so the "common denominator for these fractions is the
product x(2x+ 3): multiply both numerator and denominator of the first fraction by x to get \(\displaystyle \frac{x^2}{x(2x+ 3)}\). Multiply both numerator and denominator of the second fraction by 2x+ 3 to get \(\displaystyle \frac{2x+ 3}{x(2x+ 3)}\).
Now, we have \(\displaystyle \frac{x^2}{x(2x+ 3)}+ \frac{2x+ 3}{x(2x+ 3)}= \frac{x^2+ 2x+ 3}{x(2x+ 3)}\)
We want \(\displaystyle \frac{x^2+ 2x+ 3}{x(2x+ 3)}< 0\) which means we want the fraction to be
negative. We need to know, now, "the product or quotient of two positive numbers is positive and the product or quotient of two
negative numbers is also positive". In order to have a product or quotient of two numbers
negative, the two numbers must be of
opposite sign. That is, one positive and the other negative.
As we have already seen, the numerator of that fraction is always positive. In order for the fraction to be negative, the denominator must be negative: we must have x(2x+ 3) negative. Since that is the product of x and 2x+ 3,
they must be of different signs: we must have
either "x> 0 and 2x+ 3< 0"
or "x< 0 and 2x+ 3> 0".
1) x> 0 and 2x+ 3< 0. If 2x+ 3< 0 then 2x< -3 so x< -3/2. But then "x> 0" is not true. This case is impossible.
2) x< 0 and 2x+ 3> 0. If 2x+ 3> 0 then 2x> -3 so x> -3/2. But we also have x< 0 so we must have x< 0 so -3/2< x< 0.
is always positive for all real values of x. Hence find the range of values of x that satisfies
.
for all real values of x,
, for all real values of x.
,
.