Find the range of values of letter a for which the product (5+a)(3–a) is positive.

Question

 This is a question that I don't know that's on my RSM homework

 

Answer 1

Let's start by asking an easier question: for what values of a does (5+a)(3-a) = 0? There's only two possibilities, once when 5+a=0  (meaning a=-5) and another when 3-a=0 (meaning a=3). We can use this to create three separate regions:

 

1. a < -5
2. -5 < a < 3
3. 3 < a

 

That's great, but how can we use this information? In order for the product to switch from positive to negative (or vice versa), the product must be zero at that point. Thus, we only have to compute the product once in each region. If the product is negative, it will remain negative in the entire region since it cannot cross the zero-line.

 

As an example, let's use the following values of a taken from each region and find the corresponding product:

 

1. -6: (5 + (-6))(3 - (-6)) = (-1)(9) = -9
2. 0: (5 + (0))(3 - (0)) = (5)(3) = 15
3. 4: (5 + (4))(3 - (4)) = (9)(-1) = -9

 

Only region 2 has a positive value, so the product (5+a)(3-a) only has a positive value when -5 < a < 3.