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Simplifying versus Solving (page 3 of 3)

Related to these simplification problems are some solving problems. For instance:

• Solve 3 + 2[4x – (4 + 3x)] = –1

As usual, I'll simplify from the inside out:

      3 + 2[4x – (4 + 3x)] = –1
    3 + 2[4x – 1(4 + 3x)] = –1
3 + 2[4x – 1(4) – 1(3x)] = –1
         3 + 2[4x – 4 – 3x] = –1
                 3 + 2[1x – 4] = –1
          3 + 2[1x] + 2[–4] = –1
                     3 + 2x – 8 = –1
                     2x + 3 – 8 = –1
                           2x – 5 = –1
                     2x – 5 + 5 = –1 + 5
                                 2x = 4
                                   x = 2

It is not required that you write out this many steps and, once you get comfortable with the process, you'll probably do a lot of this in your head. But until you reach that comfort zone, you should write things out this clearly and completely. Always remember, by the way, that you can check your answers in "solving" problems by plugging them back in to the original equation.  In this case:

3 + 2[4x – (4 + 3x)]

= 3 + 2[4(2) – (4 + 3(2))]   <==(I've plugged in "2" for "x")
= 3 + 2[8 – (4 + 6)]
= 3 + 2[8 – (10)]
= 3 + 2[–2]
= 3 – 4
= –1

Since this matches the original equation, we know that "x = 2" is the correct solution.

Note the difference between this last problem and all the preceeding ones. This was a "solving" problem, whereas the previous ones had been "simplifying" problems. That is, this last problem started with an equation (something with an "equals" sign in it) and I was supposed to find a solution (something of the form "(variable) equals (some number)"). On the other hand, for the previous problems I had started with an expression (with no "equals" sign in it) and ended up with a different version of the same expression (still with no "equals" sign in it).

Why am I making a big deal about this? Because many students, for some reason, conflate equations and expressions. That is, given an expression to simplify, the student will somehow turn the problem into an equation to solve. This error will look something like this:

• Simplify 6 – 4(2x + 3) + 10x

As you can see, the student somehow (at the arrow) converted the expression "2x – 6" into the equation "2x = 6", and "solved" the expression. But the original problem didn't have an "equals" in it, so there was nothing to solve. Be careful that you don't do this.

Let's do some more problems:

• Simplify 4 – 10[x + (2x – 3)] + 12x

This is a "simplification" problem, since there is no "equals" sign in the original problem. I will simplify the expression from the inside out, and will end up with an equivalent expression as my answer.

4 – 10[x + (2x – 3)] + 12x
4 – 10[x + 2x – 3] + 12x
4 – 10[3x – 3] + 12x
4 – 10[3x] – 10[–3] + 12x
4 – 30x + 30 + 12x
–30x + 12x + 4 + 30
–18x + 34

• Solve 2(x + 3) = 4 – (2 – x)

This is a "solving" problem, since there is an "equals" sign in the original problem. I will work on both sides of the equation, and will end up with a solution of the form "(variable) equals (some number)" as my answer. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

     2(x + 3) = 4 – (2 – x)
2(x) + 2(3) = 4 – 1(2) – 1(–x)
       2x + 6 = 4 – 2 + 1x
       2x + 6 = 2 + x
2x – x + 6 = 2 + x – x
         x + 6 = 2
   x + 6 – 6 = 2 – 6
               x = –4

Let me stress once again: To do "simplification" problems successfully, you need to take the time to write out each step. Work from the inside out, and be careful with the "minus" signs. Don't forget the Order of Operations, and don't make the mistake of confusing "simplifying" with "solving".

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