Solve for X

Date: 10/09/1999 at 15:52:22
From: Jenna 
Subject: I don't understand 

Dear Dr. Math,

I'm so confused! I have my math book in front of me and I don't 
understand how to do a problem like this:

     4x - 1/2 = -3/2

Could you help me and explain it so that I can understand? My book 
just confuses me.

Date: 11/03/1999 at 21:55:22
From: Doctor Sandi
Subject: Re: I don't understand 

Hi Jenna,

Don't worry, we'll get to the bottom of it and work it out. Math books 
can be pretty confusing because they use math language, which is a 
whole new concept in itself that needs to be learned along the way and 
makes things seem confusing. I would encourage you to persevere with 
your math book though - even though it is confusing, if you keep on 
reading it and trying to follow the examples in the book, it will get 
easier to understand the more you read it. And, as you get into higher 
grades with math, if you can understand a math book, believe me, 
you'll do a lot better than you would otherwise.

What I will do, Jenna, is make up an equation similar to yours, and 
we'll work through it, find the answer, and then you'll be able to use 
the same process to find the answer to your problem on your own. But 
before I go on, let me say something about the way you wrote your 
question. You need to put brackets around the fractions; otherwise 
someone might think it could mean (4x-1)/2 = (-3/2). I'm hoping that 
I've guessed right though, and have interpreted your question 
correctly.

Let's work through this one:

     2x - (1/2) = (-5/2)

We need to find the value of x, so we have to get the 2x on its own. 
The first thing to do is to take the -(1/2) over to the other side. Do 
you remember that when you take a negative number over to the other 
side of an equation, it becomes a positive number? If the number is 
negative, to move it to the other side of the equation, the negative 
needs to be "undone" and it will then become a positive. So then we 
have:

     2x = (-5/2) + (1/2)

You could work this out on your calculator, but since they both have 
the same denominator (number on the bottom of the fraction) you could 
say -5 + 1 = -4, so 2x = -4/2, which cancels down to 2x = -2. 

Now let's have a look at the x again. It is nearly on its own but not 
quite. We need to get rid of the 2 that is multiplying the x. We need 
to move it to the other side and "undo" the multiplying action. What 
do we need to do with the 2 when we apply it to the other side to 
"undo" the multiplication? Yes, you're right, we'll divide. Then we 
have:

     x = -2/2
       = -1

There, it wasn't really that hard in the end when you worked through 
it with me was it? I hope not, anyway. But if it was, take heart 
because EVERYTHING is hard to do when we first learn how to do it. So, 
my advice to you would be to do as many of these kinds of equations as 
you possibly can, and the more you do, the easier it will get, I 
promise.

Something that also helps immensely is that now that you have found 
the value of x, you could put it back into the original equation to 
see if it all checks out. Then you know that you're right (or wrong). 
We'll do it with this one:

     2x - (1/2) = (-5/2)

So if we write

     2(-1)-(1/2) = (-5/2)

we have put the -1 in place of the x and we're hoping that when we 
work through it that it does indeed solve to be (-5/2).

Okay, 2 times -1 = -2. -2 - (1/2) = -(5/2). Yes! Now you know that the 
answer that we found for x was right, because when we put x = -1 back 
into the equation and solve it we get -(5/2) as the answer, just as 
it's meant to be. This is good to remember in exams (if you have 
time). You can often check your answers by working backward.

Here are some links that you might like to have a look at. You'll see 
that they are questions that other people have asked Dr Math together 
with the answers that they received.

   Algebra
   http://mathforum.org/dr.math/tocs/algebra.middle.html   

   Equations
   http://mathforum.org/dr.math/tocs/equation.middle.html   

   Negative Numbers
   http://mathforum.org/dr.math/tocs/negative.middle.html   

   Word Problems
   http://mathforum.org/dr.math/tocs/wordproblem.middle.html   

If you click on "Back to Middle School Level" on these pages you'll 
see that there are many more topics, and if you click on "Search Dr 
Math" you'll be able to search the whole archive for whatever topic 
you want.

In the meantime though, if you don't understand how we did the 
equation above and you have more questions about this type of thing or 
any questions about any other areas of mathematics that you would like 
to ask, we'd like to hear from you. Just email us and we'll try to 
help.

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