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Expert: - 12/25/2008


Question
Hi,
I'm working on a math course in college. I can't seem to understand how one type of linear inequality works. For example:
-2y + 4 > -6/7y - 3

I've been using the Aleks software as provided by my college, and I do solve some problems at times, but I'm so confused on which I need to move to the other side (ie. the -2y to the right side of the equation or the -6/7y to the left. What also confuses me is when I should change the sign from a "<" to ">" and vice versa. I'd really appreciate your help a lot.
Thanks in advance.

Answer
When presented with this type of problem, the first thing to do is put the variables on one side and the constants on the other.

Add 6/7 y to both sides and subtract 4 from both sides.

You get -8/7 y > -7.

In the reason to change the sign, consider 4<7.  If we multiply by a negative, we get -4 and -7.  It can be seen that -4 is greater than
-7, so the inequality needs to be changed to -4 > -7.

The last operation done here is to multiply by -7/8, giving
y < 49/8.

Another to see that the sign change was necessary is to look at a plot of the soluton to both equations.  For both equations, it can be seen that if y=49/8, then we have an equality.

A good way to check and see which side should be chosen is to put in y = 0.

The number 0 is greater than -7, so this should be in the shaded region, so the > is correct.

The number 0 is less than 49/8, so we can see that 0 is in the right place and that the sign is correct.

When faced with problems like this, I've found it good to remember something simple,

The number 1 is less than 2,
  but -2 is less than -1
so when you wonder what to do,
  go change the sign - and now you're done.

So remember, only change the sign when multplying by a negative.

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