The answers to the True-False quiz at the end of the last post are all false. This brings us to absolute value, another topic I want to concentrate on for my upcoming Algebra 1 class. Absolute value becomes a concern in calculus too which I will discuss as the last example below.

There a several “definitions” of absolute value that I’ve seen over the years which I mostly do not like

- The number without a sign – awful: all numbers except zero have a sign
- The distance from zero on the number line – true, but not too useful especially with variables
- The larger of a number and its opposite – true, but not to useful with variables

So I propose to give them an algorithm: If the number is positive, then the absolute value is the same number; if the number is negative then its absolute value is its opposite. Of course this is really the definition.

So I’ll soon express this in symbols

$latex \left| a \right|=\left\{ \begin{matrix}

a & \text{if }a\ge 0 \\

-a & \text{if }a<0 \\

\end{matrix} \right.$

Now interestingly this is probably the first piecewise defined function an Algebra 1 student may see, or at least the first one that’s not artificial. So this is a good place to start talking about piecewise defined function and the importance of talking about the domain. And of course we’ll have to take a look at the graph.

Sometime we will have to start solving equations and inequalities with absolute values. So here is the next thing I understand but do not like and will try to avoid. Solve the equation: , Answer including work: . But I think a longer way around is also better:

If then so or if , then Solution: or .

Longer? Sure. I hope that by making the students write that a few times that when they get to solving that it will be natural to say

If then so or if , then Solution: or

The last case may take a little more discussion. Solve . Starting the same way

If then so which really means

if , then which really means . Then the union of these two sets looks like an intersection. The solution is

Quite often the equation and the two types of inequalities are treated as separate problems: with = you go with on the other side, with > you have a union pointing away from the origin and with < you have somehow an intersection. Who needs to remember all that when this idea works all the time?

Example: Solve

If , then and , so and or more precisely $latex \tfrac{1}{2}

If , then and , so and or more precisely . The union again becomes an intersection and the answer is

Finally an example from calculus. On the 2008 AB exam, question 5 asked student to find the particular solution of a differential equation with the initial condition . After separating the variables, integrating, including the “+*C*” and substituting the initial condition students arrived at this equation which they now need to solve for *y*:

How can you lose the absolute value sign? Simple, near the initial condition where *y* = 0, so replace with and then go ahead and solve for *y*

I don’t think I’ll try this one in Algebra 1, but maybe it will come in handy when they get to calculus.