Inequalities: Add or Subtract?

Date: 10/21/2001 at 11:19:41
From: Sam
Subject: Inequalities, add 1 or subtract?

Dr. Math:

Sometimes, when doing inequalities problems, I have to add or subtract 
one from the answer that I have calculated. I just don't get it - 
when to add, subtract, or do nothing at all!

Such as in one such question:

The Social Services Group of a certain Secondary School is organizing 
a fund-raising concert. Teachers and students can buy tickets to 
attend the concert. Each teacher ticket and each student ticket is 
$200 and $100 respectively. The organizing committee wants to raise 
more than $100,000. If 50 teachers have bought tickets to the concert, 
at least how many students need to buy in order to meet the target?

Here are my steps:

Let x be the least no. of students:

200 * 50 + 100x > 100,000
100x + 10,000 - 10,000 > 100,000 - 10,000
100x > 90,000
100x / 100 > 90,000 / 100
x > 900

901 (I don't know why, only because teacher says so) students need to 
buy the tickets.

Can you please tell me why?
Sam

Date: 10/22/2001 at 14:33:59
From: Doctor Peterson
Subject: Re: Inequalities, add 1 or subtract?

Hi, Sam.

It sounds like the part that gives you trouble is translating back 
from an inequality like x > 900 (meaning "x is MORE THAN 900") to a 
word form like 'x is AT LEAST 901'.

Picture the number line:

                           o======================>
    <--+---+---+---+---+---+---+---+---+---+---+-->
      895 896 897 898 899 900 901 902 903 904 905

Any number greater than 900 will work, but 900 itself will not (since 
900 students would give exactly $100,000, not "more than" that much). 
Since only whole numbers of students are allowed, the smallest number 
possible is 901, the next whole number AFTER 900.

So you add one in order to go from the 900, which is not large enough, 
to the NEXT number, which is. There must be AT LEAST 901 students in 
order to have MORE THAN 900.

Of course, the problem was deliberately set up to force you to do 
this. Normally, the target would more likely be "at least $100,000" 
and the answer would then be "at least 900 students."

Let me know if you need any more help with this.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/