Foundations of Algebra II

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Welcome to the new season of 12 and Beyond’s math-based series, It All Adds Up! As I embark on my sophomore year of high school, I’ve begun taking Algebra II. Already, in comparison to Geometry, I feel the course is much harder, and the pace is insane- just how I like it.

So far, we’ve jumped all over the place in terms of topics, since we essentially covered 2 1/2 chapters of algebra review in a week. Thus, I’m going to pick and choose a few interesting topics to discuss today: Absolute Value Equations and Interval Notation. 

In terms of basic algebra foundations, Kahn Academy has some excellent courses in terms of Algebra I. In order to attempt to stay relatively caught up, I’ll be skipping over much of that, if just to focus on major concepts. Thus, throughout the year, lessons are going to seem all over the place. Remember, this isn’t a textbook, and is meant to help me further enforce my understanding of mathematics.

This’ll be a fun year! Let’s begin.

Absolute Value Equations

|6x + 6| = 36

Seems simple enough, right? Given the principle of absolute value, think of the bars as parenthesis, except once the inside of the parentheses is solved, it must be positive. This means that x can potentially be negative, so what we can do is set up two separate equations:

6x + 6 = 36    &    6x + 6 = -36

With the second equation, x will turn out to be a negative (-5), but since the variable is found inside the AV bars in the original equation, the result will be a positive 36. After solving each of these equations using basic algebra, they can both serve as solutions.

With a few exceptions.

No Solution

If an absolute value will always be positive, than it can’t possibly be equal to a negative. If the right side of the equation is negative, we can automatically determine that there is no solution.

Extraneous Solutions

While a negative variable may be fine in certain cases, when a variable is found (perhaps as a coefficient) on the non-AV side of the equation, there’s a potential that the result could be similar to a no solution problem, where one (or both) solutions may not be valid. When you see something like |6x + 7| = 4x, make sure to plug in both of your solutions into the original equation (WITH the AV bars- and not a reciprocated right side) and ensure the solution is correct.

All Real Numbers

In the past, this may have been referred to as infinitely many solutions. When an AV equation results in 0=0 or # = #, then any real number could be inserted.

Absolute Value Inequalities

|6x + 6| > 36

We treat these the same way in separating them into two separate inequalities, except for one key difference – when we negate one side, we also have to flip the inequality symbol.

6x + 6 > 36    &    6x + 6 < -36

These can be solved in the same way, with prior knowledge of inequality solving. Additionally, once each is solved, graphing them should be pretty easy. Based on the range of numbers, they can be classified as an and or or equation. As you know, the classification determines which way the graph points (and if there are multiple rays, or just one segment).

No Solution

Similarly, an absolute value cannot be less than a negative, so if a negative is found on the right side of the inequality (without a variable) then we can deem it to have no solution (though if you have time, it’s worth trying it). If there is a variable placed next to a coefficient on the right side (regardless of negative/positive), then be sure once again to check for extraneous solutions.

All Real Numbers

If an equation states that an absolute value is greater than a negative number, than any possible real number can be a solution (a positive will always be greater than a negative).

Integer Notation

Integer notation is a way of expressing the range of numbers classified as solutions by an inequality. Oftentimes, this involves infinities (as with an or inequality, where two rays extend endlessly). It’s written as an ordered pair, with each extent following the order of a number line. Think of it as everything from x to y.

Based on the type of inequality, Interval Notation is expressed differently:

  • With greater than or lesser than inequalities, parenthesis are used to indicate the endpoint is not a part of the solution set. This is always used adjacent to any infinity value, as there is no finite end.
  • With greater than or equal to OR lesser than or equal to inequalities, we use brackets [ ] to express the set.

You might have both a parenthesis and a bracket in one notation.

6 < x >/ 9

Treat >/ as greater than or equal to. In this case, we have an “and” inequality. Therefore, the solution set is in between these two numbers. Based on the symbols adjacent to each value, our integer notation would look like this:

( 6 , 9 ]

The lower number always comes first.

What about this or equation?

x >/ 9 or x < 7

For or equations that don’t encapsulate all real numbers, we separate two ordered pairs (in order of which falls first on a number line) with a symbol that looks like a U.

Our integer notation would be:

( -∞ , 7 ) U [ 9 , ∞ )

When the ray extends to the left (negative) side of a number line, we use a negative infinity, and vice versa. Like I mentioned, the negative infinity pair will always come first (or that closest to the negative).

Though these sections might not tie in significantly later on in Algebra II, they are relevant to what we are learning in class right now. For further extensions, this is what else I’ve been working on:

  • Direct Variation
  • Scatter Plots & Lines of Best Fit
  • Absolute Value Graphs

Thanks so much for reading! See you next month!

Gabe

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