explain these simple inequality division problems

Question

My Question: How do I know when to flip the symbol (during multiplication or division) as opposed to when to subtract/add in order to change the positions of the numbers and variables?

 

Observe:

When I did problem #4):

 

 

-162 < 18r

 

I simply divided both sides by 18 and flipped the symbol to get r> -9.

 

HOWEVER, in problem #5):

 

92 is greater than or equal to -4p

 

I divided both sides by -4 and got the incorrect answer of P is less than or equal to -23 after I flipped the sign. The answer key told me I was wrong.

After I googled this problem, I learned that I had to subtract 92 from both sides and then put 4p onto the opposite side to end up with 4p is greater than or equal to -92 and then solve. 

the correct answer was p is greater than or equal to -23.

 

My Question:

Why did I have to solve it differently in two look-a-like problems? Is there a rule to this? How will I know when to do what?

Answer 1

Well, let’s clarify/correct some of your statements, then, state the rules to follow.

For problem #4), you said:
     “  -162 < 18r
       “I simply divided both sides by 18 and flipped the symbol to get r> -9.”
       This is:       -163 < 18r
                               -9 < r       (divided both sides by +18)
     You did not “flip” the sign; but reading right to left, r > -9.
 
Then, problem #5)
          92 >= -4p
         -23 <= p        (divided both sides by -4)
    Wait!! Reading right to left, this is exactly p >= -23    (the correct answer)

There are two rules to remember:

     1. Any number may be added to (or subtracted from) both sides without changing the inequality.

     2. Multiplication (or division) by a positive number does not change the inequality;  multiplication (or division) by a negative number reverses the sense of the inequality.

Note: that “reverses the sense of the inequality” does not ever change the “=”.

That second rule is a little tricky when we don’t know whether a variable (for example, p) is positive or negative.  We usually just isolate the variable from the constants (and, sometimes we read the answer right-to-left).

This is why waiting on uncertain multiply (or divide) operations is a good idea.

Answer 2

92≥-4p

divide both sides by -4, which changes the sign

92/(-4)≥(-4p)/(-4)

-23≤p

-23≤p implies p≥-23

you don't have to make any changes like you mentioned at the beginning of the problem; wait until you are finished and then make the change