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 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 frequently when 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 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: a summary

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: a summary

Arithmetic Sequences and Series: a summary

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: a summary

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)?

Domain, Range, and Co-domain

When working with quantitative relationships, three concepts help “set the stage” in your thinking as you seek to understand the relationship’s behavior: domain, range, and co-domain.

Domain

The “domain” of a function or relation is:

  • the set of all values for which it can be evaluated
  • the set of  allowable “input” values
  • the values along the horizontal axis for which a point can be plotted along the vertical axis

For example, the following functions can be evaluated for any value of  “x”:

f(x)=2x+1\\*~\\*g(x)=x^2+5

therefore their domains will be “the set of all real numbers”.

The following functions cannot be evaluated for all values of “x”, leading to restrictions on their Domains – as listed to the right of each one:

h(x)=\dfrac{1}{x}~~~~~~~~~\text{x cannot be zero}\\*~\\*j(x)=\dfrac{1}{(x-2)(x+4)}~~~~~~\text{x cannot be 2 or -4}\\*~\\*k(x)=3x+2,~1<x<10~~~~\text{only values between -1 and 10 may be used for x}

The values for which a function or relation cannot be Continue reading Domain, Range, and Co-domain

Roots and Rational Exponents: a summary

Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are the same value):

2^3 \ne 3^2

Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
– roots, to undo exponents
logarithms, to undo bases

Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many versions of the “root” function: one for each exponent value to be undone.

Notation

The symbol for a root is \sqrt{~~~~}, and is referred to as a “radical“.  It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a grouping symbol (like parentheses) to Continue reading Roots and Rational Exponents: a summary

Logarithms: a summary

Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in the exponent (unless, of course, they are the same value):

3^2 \ne 2^3

The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent (a root), and a different one to undo the base (a logarithm).

Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.),  there are also many versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.

Inverse Functions: Logarithms & Exponentials

Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Continue reading Logarithms: a summary

Systems Of Linear Equations: a summary

What is a “system” of linear equations?

A “system of linear equations” means two or more linear equations that must all be true at the same time.

When represented symbolically, a system of equations will usually have some sort of grouping symbol to one side of them, such as the curly brace below, which is intended to convey that the set of equations should be considered all at once. For example:

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

When graphed, all of the equations in a system will be shown on the same set of axes, so that they can be compared to one another easily:

What is “a solution” to a linear system?

A solution to a system of linear equations is Continue reading Systems Of Linear Equations: a summary

Completing the Square Procedures

I have seen three approaches to “Completing the Square”, as shown below. Each successfully converts a quadratic equation into vertex form.  Which do you prefer, and why?

First Approach

This approach can only be used when you are working with an equation. It moves all terms that are not part of a perfect square to the other side of the equation to get them out of the way:

y=2x^2+12x+10

y-10=2x^2+12x

\frac{1}{2} y-5=x^2+6x

\frac{1}{2} y-5+(\frac{6}{2})^2=x^2+6x+(\frac{6}{2})^2

\frac{1}{2}y-5+9=x^2+6x+3^2

\frac{1}{2}y+4=(x+3)^2

\frac{1}{2}y=(x+3)^2-4

y=2(x+3)^2-8

Second Approach

Continue reading Completing the Square Procedures

Quadratic Equations: a summary

Adding a squared term to a linear expression, creating a quadratic expression in the process, seems like a relatively small change:

3x+2

x^2+3x+2

Yet, if this new term is part of an equation, the procedures that worked nicely when solving linear equations don’t work so well any more. Investigating what happens in such situations is useful, and leads to some new concepts and procedures.

If a quadratic equation is approached in the same way as a linear equation, it can sometimes be solved quickly:

18=2x^2\\*~\\*9=x^2\\*~\\*\pm3=x

Familiarity with square roots and how to solve linear equations are enough to solve this equation. By getting the variable all by itself on one side, the two possible solutions to the original equation are left on the other.

However, the following equation cannot be solved in the same way:

0=x^2+2x+1\\*~\\*-1=x^2+2x\\*~\\*\dfrac{-1}{x}=x+2~~~~~~~~~(a)\\*~\\*-1=(x)(x+2)~~~(b)

This equation has two “x” terms, and they are not “like terms” since one has the variable to the first power and the other to the second power. If a linear equation approach is used, moving the constant term to the other side, two un-like terms are left on the right… but what to do from here?

Continue reading Quadratic Equations: a summary