Algebra's Basic Concepts - Special Cases: Infinite Solutions And No Solutions

The typical 1st year Algebra course covers methods for solving single-variable equations, linear equations, and quadratic equations. In this article, we will review the methods for solving single-variable equations and take a look at two "special cases" that can happen when solving this type of equation. These "special cases" often cause students a great deal of confusion.
Before looking at those special cases, let's review the basic technique for solving single-variable equations. Remember: "isolate the variable" or "un-do" and reverse PEMDAS.
Example 1: 3x - 5 = 4 First, we need to add 5 to both sides.
3x - 5 + 5 = 4 + 5 Now, simplify.
3x = 9 Divide both sides by 3.
(3x)/3 = 9/3 Simplify.
x = 3
Now, check to see if this value makes the equation true. Substitute this "answer" into the original equation.
3(3) - 5? 4
9 - 5? 4
4 = 4 It checks. Thus, our solution is x = 3 and its graph is a solid dot at 3 on a number line labeled x.
Example 2: a/3 - a/4 = 8 Equations with fractions are not as difficult as most people think. The key is to eliminate the denominators. We decide what number all the denominators will divide into evenly. For 3 and 4, that number is 12. We now need to multiply every term by 12.
12(a/3) - 12(a/4) = 12(8) The denominators each divide evenly into 12, so they are gone.
4a - 3a = 96 Combine like terms.
a = 96
Now check: 96/3 - 96/4? 8
32 - 24? 8
8 = 8 It checks. So the solution is a = 96 and its graph is a solid dot at 96 on a number line labeled a.
Now, let's look at one "special case"
Example 3: 8(2x - 3) = 4(4x - 8) We need to eliminate the parentheses using the distributive property.
16x - 24 = 16x - 32 Now, move the x's to the left.
16x - 16x - 24 = 16x -16x - 32 Simplify.
-24 = -32 Well, this certainly doesn't look right! Our 16x terms cancelled each other out leaving a really weird situation. What makes it especially weird is the fact that the statement is FALSE. Did we make a mistake? No.
This is the first of those "special cases." When terms drop away and the remaining statement is FALSE, it means that the original "equation" isn't true. There is NO value that can ever make this "equation" TRUE. So this is a NO solution situation.
For this next example I chose a problem with an inequality symbol which I know we haven't studied yet. There are times when inequalities can be problematic, but this is not one of those. So we just bring the symbol down as we do with an = symbol.
Example 4: -3(x - 3) > 5 - 3x Eliminate the parentheses.(Distribute)
-3x + 9 > 5 - 3x Now, take x's to the left.
-3x + 3x + 9 > 5 -3x + 3x Simplify
9 > 5 (This is read as "9 is greater than 5.")
Well, it happened again except for one thing. The x terms cancelled out, but the remaining statement is TRUE. Since 9 > 5 is true all the time, it means the original is true all the time as well. It will be true no matter what value of x you check. Try some. Everything works! Really weird! Unlike the other "special case" that had NO solution, this time we have many solutions. This is the INFINITE solution situation.

To summarize: When solving single-variable equations, one of three possibilities will happen:
1) You will get ONE solution. x = some number
The original equation is called a conditional equation because it is true for only a certain value.
2) You will get NO solutions as evidenced by a final FALSE statement.
This original equation is call a contradiction because it can't be true.
3) You will get INFINITE solutions as evidenced by a final TRUE statement.
This original equation is called an identity because it is always TRUE.
I think you can understand why so many high school students get confused. These special cases just seem really weird!

Shirley Slick, "The Slick Tips