writing sentence as an inequality

Question

how to write

a.)  " A number n is less than or equal to -7 or greater than 12" as an inequality

 

b.) " A number p is less than 6 and greater than 2

Answer 1

Just translate directly from "English" to "math" - 

n ≤ -7 or n > 12

Now, to pretty this up, I am going to flip the second part around:

 

n ≤ -7 or 12 < n

 

Finally, I am going to replace "or" with the "intersection" sign, to make it more formally mathematical:

n ≤ -7 ∩ 12 < n

 

Similarly,

p < 6 and p > 2

I want my little alligator mouths to open the same way, so I do the trick again:

p < 6 and 2 < p

 

The trouble now is that I want the smaller numbers on the left, and the bigger ones on the right, so I move the components of the compound inequality around:

 

2 < p and p < 6

Since "p" is in the middle, I do not have to repeat it; my compound inequality is just this:

2 < p < 6