Solving Absolute Value Equations

What Does Absolute Value Mean?

The term “Absolute Value” refers to the magnitude of a quantity without regard to sign, in other words, its distance from zero expressed as a positive number. The notation used to indicate absolute value is a pair of vertical bars surrounding the quantity, sort of like a straight set of parentheses. These bars mean: evaluate what is inside and, if the final result (once the entire expression inside the absolute value signs has been evaluated) is negative, change its sign to make it positive and drop the bars; if the final result inside the bars is zero or positive, you may drop the bars without making any changes:

\lvert ~1-4~ \rvert\\*~\\*=~\lvert ~-3~ \rvert\\*~\\*=~3

Another example is:

\lvert ~4-1~ \rvert\\*~\\*=~\lvert ~3~ \rvert\\*~\\*=~3

Note that absolute value signs do not instruct you to make “all” quantities inside them positive. Only the final result, after evaluating the entire expression inside the absolute value signs, should be made positive.

\lvert ~1-4~ \rvert~\ne~\lvert ~1+4~\rvert~~\text{ Do not make this mistake!}

Absolute Value expressions that contain variables

Just as with parentheses, absolute value symbols serve as grouping symbols: the expression inside the bars must be evaluated and expressed as either Continue reading Solving Absolute Value Equations

Combining or Collecting Like Terms

The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.

Quantities With Units

Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?

Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?

The quantities of apples may be combined because addition or subtraction only work with  Continue reading Combining or Collecting Like Terms

Simplifying Fractions

Consider the following fraction… can it be simplified? If so, why can it be simplified?

\dfrac{12}{15}

The above fraction can be simplified, because both numerator (the top expression) and denominator (the bottom expression) share a common factor. By factoring both numerator and denominator, then pulling out the factor they each have in common (using the reverse of how fractions are multiplied), we end up with a fraction that equals one being multiplied by a simpler fraction than the original:

\dfrac{12}{15}~~=~~\dfrac{3\cdot 4}{3\cdot 5}~~=~~\dfrac{3}{3}\cdot\dfrac{4}{5}~~=~~1\cdot\dfrac{4}{5}~~=~~\dfrac{4}{5}

Since multiplying by one does not change a quantity, we can drop the multiplication by one from our expression, leaving behind the simplified fraction as our final expression on the right.

This is the only process that allows you to Continue reading Simplifying Fractions

Negative Differences

Algebra is a set of rules that allow us to change the appearance of an expression without changing the quantitative relationship that it represents. Sometimes the changes in appearance are greater than expected, causing us to doubt whether two expressions really do represent the same quantitative relationship.  The ways in which negative differences can be rewritten seem to surprise people until they become accustomed to them.

Consider a difference that is being subtracted:

b-(a-3)

If we wish to eventually drop the parentheses, we’ll have to distribute the negative sign in front of them first.  Leaving the parentheses in place while Continue reading Negative Differences

Negative Fractions

Where should I put the negative sign when I am writing a fraction like negative two thirds?  As long as you write only one negative sign, it does not matter where you put it.

Subtraction is the same thing as the addition of a negative. The negative of a number is created by multiplying the number by negative one. These rules apply to fractions as well, so:

-\dfrac{3}{5}\\*~\\*~\\*=(-1)(\dfrac{3}{5})\\*~\\*~\\*=(\dfrac{-1}{~1})(\dfrac{3}{5})\\*~\\*~\\*=\dfrac{-3}{~5}

So, placing the negative sign before the entire fraction (subtracting the fraction) is equivalent to Continue reading Negative Fractions

Geometric Sequences and Series

Geometric Sequences / Progressions

Pick a number, any number, and write it down.  For example:

5

Now pick a second number, any number (I’ll choose 3), multiply it by the first number, then write their product down to the right of the first number:

5,~15

Now, continue multiplying the second number by previous product and writing the result down… over, and over, and over:

5,~15,~45,~135,~405,~1,215, ...

By following this process, you have created a “Geometric Sequence” or “Geometric Progression”, a sequence of numbers in which the ratio of every two successive terms is the same.

Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence or progression. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

a_1

This notation is  Continue reading Geometric Sequences and Series

Arithmetic Sequences and Series

Arithmetic Sequences

First, a terminology aside: the terms “sequence” and “progression” are interchangeable. An “arithmetic sequence” is the same thing as an “arithmetic progression”. From here on in this post, the term “sequence” will be used… but if you live in a place that tends to use the word “progression” instead, it means exactly the same thing. So, let’s investigate how to create an arithmetic sequence (also known as an arithmetic progression).

Pick a number, any number, and write it down.  For example:

5

Now pick a second number, any number (I’ll choose 3), add it to the first number, then write their sum down to the right of the first number:

5,~8

Now, continue adding the second number to the sum and writing the result down… over, and over, and over:

5,~8,~11,~14,~17,~20,~23,~26,~29, ...

By following this process, you have created an “Arithmetic Sequence”, a sequence of numbers that are all the same distance apart when graphed on a number line:

ArithSequence

Vocabulary and Notation

In the example above 5 is the first term, or starting term, of the sequence. To refer to the starting term of a sequence in a generic way that applies to any sequence, mathematicians use the notation

a_1

This notation is Continue reading Arithmetic Sequences and Series

Piecewise Functions and Relations

While many relationships in our world can be described using a single mathematical function or relation, there are also many that require either more or less than what one equation describes.  The behavior being described might start at a specific time, or its nature changes at one or more points in time. Two examples of such situations could be:

Piecewise1
Acceleration up to a speed limit
Piecewise2
Free fall then controlled descent

In the graph on the left, note that the blue line starts at the origin. It does not appear to the left of the origin at all. Furthermore, when x = 3 the blue line stops and the green line begins – but with a different slope.

In the graph on the right, note that the blue curve starts at x = 0.  It does not appear of the left of the vertical axis at all.  And when x = 3 the blue parabola turns into a green line with a very different slope. And the green line stops at x = 5.5, just as it reaches the horizontal axis.

These graphs do not seem to follow all the rules you were taught for graphing lines or parabolas. Instead of being defined over all Real values of x, they start and stop at specific values. The graphs also show (in this case) two very different functions, but in a way that makes them look as though they are meant to represent a single, more complex function.  Both of these graphs are Continue reading Piecewise Functions and Relations

Linear Systems: Why Does Linear Combination Work (Graphically)?

A system of linear equations consists of multiple linear equations.  The solution to a linear system, if one exists, is usually the point that all of the equations have in common. Occasionally, the solution will be a set of points.

There are four commonly used tools for solving linear systems: graphing, substitution, linear combination, and matrices. Each has its own advantages and disadvantages in various situations, however I often wondered about why the linear combination approach works. My earlier post explains why it works from an algebraic perspective. This post will try to explain why it works from a graphical perspective.

Consider the linear system:

\begin{cases}y=-3x+2\\y=x-6\end{cases}

which, when graphed, looks like: Continue reading Linear Systems: Why Does Linear Combination Work (Graphically)?

Where’s the mistake?

I have started a separate blog devoted to helping students learn to find mistakes in worked problems (their own, or someone else’s). If this is of interest, check it out:

http://mathmistakes.wordpress.com/

7/17/11 Update: There can be great value in work that contains mistakes. Learning to catch your own mistakes is a critical life skill, as is learning to review other people’s work while seeking to understand it fully (the best way to do this is by looking for mistakes).

Along these lines, I came across an interesting blog posting by Kelly O’Shea.  She came up with the idea of insisting that each group who is presenting try to sneak a mistake in their work past their peers.  Brilliant!

Algebra Intro 1: Numbers and Variables

This post begins a series intended to help introduce or re-introduce some of the core concepts of Algebra. It is often very helpful to re-visit these concepts with students who may have memorized their way through previous math courses without slowing down to contemplate some of the concepts behind Algebra.

Numbers

Numbers are used in the English language as both nouns and adjectives. However, the only physical instance of a “number” in our world is a symbol or group of symbols such as “23”. Most of the time, we seem to use numbers as adjectives: “Look at the two trees.” Continue reading Algebra Intro 1: Numbers and Variables

Algebra Intro 2: Addition

Mathematical thinking probably started with addition. Someone may have combined two piles of bricks, and wondered how many were in the single large pile. Addition is the mathematical term that describes “joining quantities together”.

Properties of Addition

Looking at the patterns that can occur when quantities are joined together, you might have noticed that it does not matter whether 2 bricks are added to a pile of 3, or 3 bricks are added to a pile of 2… either way, we still end up with a pile of 5. So, the order in which we add two quantities does not change the result. This probably makes intuitive sense to you when you visualize the situation above and the final pile that results. Continue reading Algebra Intro 2: Addition

Algebra Intro 3: Subtraction

Once addition has been explored a bit, it leads pretty naturally to a new question: if there are three bricks in a pile, how many bricks do I need to add to it so that there will be five in the pile?

Our addition problems were all phrased using a pattern like

number + number = what?

and the question above rearranges it a bit:

number + what? = number
or
what? + number = number

This question is usually asked as “What is the difference between Continue reading Algebra Intro 3: Subtraction

Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

Negative Numbers

Negative numbers are something new and interesting to think about. What do they mean?

They arose from changing the order in which we subtracted two numbers. While we usually think of a “difference” by starting our thought process with the larger number, when we fail to do that and try to Continue reading Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

Algebra Intro 5: Addition, Subtraction, and Terms

What happens when problems involve both addition and subtraction? Addition is both associative and commutative, and subtraction is neither…

One solution is to follow the order of operations (parentheses, exponents, multiplication, division, addition, subtraction) working from left to right in the event of a “tie”. This will always produce the intended result.

However, the order of operations does not provide any guidance about how Continue reading Algebra Intro 5: Addition, Subtraction, and Terms