Interpreting "Less" and "Less Than" in Word Problems

Date: 10/18/2004 at 20:49:37
From: Charles
Subject: A tough word problem that I cannot crack.

Find all numbers such that 9 less than the product of the number and
-4 is less than 7.

The confusion comes when the sentence breaks into two parts, like 
this:

9 < x = -4 < 7

Date: 10/18/2004 at 22:22:38
From: Doctor Peterson
Subject: Re: A tough word problem that I cannot crack.

Hi, Charles.

The words "greater" and "less" in a problem are often tricky, because
they can be used in different ways:

  5 is less than x     means     5 < x

  5 less than x        means     x - 5

  5 less x             means     5 - x

Your problem uses the first two!  You aren't told that "9 IS less than
...", but that "9 less than [something] IS less than 7".  Here is how 
I think through a sentence like yours:

"9 less than ..."
  We're going to be subtracting 9 from something; what is it?

"... the product of the number and -4 ..."
  There are two numbers being multiplied, namely x and -4.  That's
  what we have to start with: (x)(-4), from which we subtract 9.
  So we have

    -4x - 9

"... is less than 7"
  The expression we've just found is less than 7, so we have the
  inequality

    -4x - 9 < 7

Now you can solve it.

If you need more help, please write back and show me how far you got.


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