The Good, the Complicated, and the Unfortunate

The good: I made it to the quadratic equation before the books were due.

The complicated: I didn't get much farther. Not so much because I didn't understand it, but The complicated: I didn't get much farther.

The complicated: My arithmetical skills blow on the same level as Johann Kepler's.* If I don't rewrite the determinant as

I will blow the sign and the whole problem. Which is probably where my issues were in 10th grade. So I'm pausing to practice, catch up the blog (for real this time, I have a new toy perfectly capable of the job much quicker than computer Photoshopping). But hey, I made it this far, didn't I? That's a pile of awesomeness. I can look at a matrix and say, "What do you want me to do with that?" I don't think synthetic division relates to creative accounting. And I remember the almighty quadratic equation:

It's just going to take a little longer than I thought.

*Johann was struggling to explain a mystery of the planets using the "five perfect shapes" of Pythagorean geometry. Perfect in the same way ancient Greeks considered octaves, fifths, and fourths perfect, but have any on your theory project and you'll regret it. He struggled to find an arrangement that made sense. He actually found the correct answer...and screwed up the arithmetic. It's okay, Kepler. I know how you feel.

I am Soooooooooo Behind

The further I progress in algebra, the harder it is to blog about my progress. Not because I don't understand what I'm doing -- I do! The further I go the more complicated the notation gets, which means either lots of sloppy computer shortcuts (I hate the look of x^2) or tedious html coding. Type polynomial. Go into html editor. Add <sup> tags. Find out I didn't close half of them. Go back and close them. Rinse, lather, repeat. I don't think I can begin to handle a post on matrices without going into Photoshop, drawing one, then using it as a template for all the steps.

I can hear you now. "Laura, for reals. No one reads this blog. Why put yourself through all this?"

1. Blogging reinforces my understanding of the material. It makes me organize my thought process.
2. Consider this a public copy of my notebook, which is already beginning to fray at the ends. And fill up. Seriously, I'm almost out of paper, and I haven't finished the book!
3. Laugh at the crazy woman who fills her spare time with algebra. And then one day your kid will come home with that terrifying textbook filled with x, y, polynomials, and WORD PROBLEMS. 

And then you'll remember that nutty friend of yours on Facebook who posted about algebra and feel (only a little) bad about mocking her nerdiness while you're frantically looking for her blog. All the while your kid is sitting there with a page of unfactored polynomials.

So yes, I am going to get all these posts done and up. Eventually. I'm taking a breather before I start complex numbers. As I vaguely recall that's where Algebra II hit the fan for me. Or maybe it was functions. The point is I'm going to use this pause time to catch up the blog. There's also a large divergence starting between Idiot's Guide and Everything. I'm not yet sure if it's simply different sequencing or if they're going in completely different mathematical directions. I think it's a good place to rest. The books are due February 8, and if I don't finish there's no rule saying I can't check them out again. Well, unless there's a hold on them.

COMING SOON!!!!

See...MATRICES!

Adding, subtracting, multiplying,

and introducing

DETERMINANTS

See...POLYNOMIALS!

Adding, subtracting, multiplying, dividing,

with special guest star

FACTORING

See...RADICALS!

Adding, subtracting, multiplying, dividing,

and FRACTIONAL EXPONENTS!

Systems of Equations

In my word problem slogging I accidentally jumped ahead and worked on this without even knowing it.

A system of equations means they share the same x and y values. The equations themselves are not equal to each other. There are three ways to solve them.

Graphing: Slap the equations on a coordinate plane to find your answer.

Pros: Visual confirmation of (x,y) values.

Cons: Good luck eyeballing the difference between 4.33 and 4.4

Naturally algebra books rig it so any problems you're instructed to solve by graphing will have nice round non-decimaled numbers.


Substitution: This is my personal favorite. Solve one equation for a variable (usually whichever's easiest to isolate), plug it in to the other equation, get a numerical value for a variable, then use that value to solve for the remaining variable. I made that more confusing that is really is, didn't I? Let me demonstrate.

Solve system x - 4y = 11

                      3x + 7y = -5

Since x has no coefficient in the first equation, let's solve it for x.

x - 4y = 11

x - 4y + 4y = 11 + 4y

x = 11 + 4y

Now take x = 11 + 4y and use in the second equation. No, you can't use it in the first equation. You'll end up with 11 = 11. Even I know you can't do that. How did I know that? Never mind...

3(11 + 4y) + 7y = -5

33 + 12y + 7y = -5

33 + 19y = -5

19y = -38

y = -2

Whether you want to solve for x in the first or second equation is up to you. The first one looks easier to me.

x = 11 + 4y (Why go here instead of standard form? Do you really want to move the ys around again?)

x = 11 + 4(-2)

x = 11 + -8

x = 3

The coordinate pair for this system is (3,-2)

Elimination Method: I get it, but I don't use it unless instructed. Multiply one or both equations by a real number, set it up like an addition/subtraction problem, and eliminate a variable.

Solve system 2x - y = 13 

                     6x + 4y = 4

Greatest common factor time! You can do either 4(2x - y = 13) or -3(6x + 4y = 4). Why -3? If you go that route your goal should be to eliminate the 6x, for which you'll need it's opposite. For some odd reason I went with -3 even though I walk the path of least resistance.

 6x + 4y =    4

-6x + 3y = -39

 0x + 7y = -35

y = -5

Solve for x

2x - (-5) = 13

2x = 8

x = 4

The coordinate pair for this system is (4,-5)

Where Did My X and Y Go, and Should I Care?

Sometimes variables disappear when solving systems of equations. Check your math FIRST before classifying it under one of these two exceptions.

1. If the ending statement is FALSE (7 = 5), the system has no solutions and their lines are parallel on a graph. If you're not staring down a classroom clock, graphing the equations is a good way to confirm it. If you wind up with intersecting lines, you messed up somewhere. Please don't ask me to figure out where.

2. If the ending statement is TRUE (10 = 10), the system has multiple solutions and is inconsistent. This means their graphs overlap and the equations are part of a dependent system. They'll never break up with each other because the crazy they know seems safer than unknown crazy.

Systems of Inequalities

Both Everything and Idiot's Guide solve systems of inequalities by graphing them, shading overlapping regions, and picking test points within the answer region to confirm. Everything tells you how to plot it on a TI-81. I don't have a graphing calculator, nor do I ever plan on owning one. If I recall correctly, learning how to program those suckers is almost as frustrating as learning the math. Sorry, can't be of much help here.

So...I have an admission to make. My blogging hasn't slowed because I've stalled in the material. Quite the contrary. My blogging's stalled because I got my daughter Lego Star Wars: The Complete Saga for her birthday. And by got it for her birthday, I mean bought it for us to play together under the pretext of a birthday gift. And by play together, I mean she watches me play because her motor skills aren't good enough to handle the controls yet.

Okay, fine. I like Lego Star Wars. I've only cared about two video games in my life. This is one of them. (The other is the original 8-bit Tetris, but only if I have to blow on the cartridge.)

Monday and Tuesday I waded through matrices and Cramer's sadistic Rule. Today I got through adding, subtracting, multiplying, and dividing (both long and synthetic) polynomials. The matrix post will take some time. It requires a lot of drawing in Photoshop and formatting. All About Polynomials should be up tomorrow.

Inequalities

Yes, I did start the matrix today! But first I need to write down what I've learned about inequalities before I forget it all.


Why have inequalities at all? Here are some examples of real life inequality situations -- mathematical ones, anyway. Most of us don't set up word problems when we think through them. We just make do. I only worry how to solve what is presented to me.

Second grade (guessing) math sign review

x < = less than but not including
x > = greater than but not including
x < = less than or equal to
x > = greater than or equal to

The "but not including" part isn't usually included when one first learns about greater than and less than. It is crucial to note the difference when graphing inequalities.

Rule #1 for inequalities is reverse the greater/less than sign if you multiply or divide the entire inequality by a negative number. Or the "faster, more intense" version:

REVERSE THE GREATER/LESS THAN SIGN IF YOU MULTIPLY OR DIVIDE THE ENTIRE INEQUALITY BY A NEGATIVE NUMBER.


Basically inequalities are solved the same way as equalities but without the equals sign.

Solve -5x + 3 > -32 for x.

-5x + 3 > -32
-5x + 3 - 3 > -32 + -3
-5x > -35
-x > -7
-1(-x > -7) -- Apply Rule #1
x < 7

Meaning x is any number less than but not including 7. 6.99 will work. 7 won't.





Compound Inequalities

Using only one sign isn't enough? You want to x to be greater than AND less than in the same inequality? Compound equations are for you. Consider it minimum and maximum speeds on the interstate. 45 < x < 70 where x is ticket-free speed. In algebra books x has a coefficient which must be dealt with.

Solve inequality -4 < 3x + 2 < 20

1. Isolate x.

-4 + (-2) < 3x + 2 + (-2) < 20 + (-2)

-6 < 3x < 18

2. Eliminate the coefficient.

(-6)/3 < (3x)/3 < 18/3

-2 < x < 6

On this hypothetical highway you can drive as low as and including -2mph and up to but not including 6mph.

Solving Absolute Value Inequalities

Absolute value is the Jessica Fletcher of algebra. Everything is nice, neat, and tidy in your police report head. Then here comes this well-intentioned mystery writer asking too many questions two little bars which upend all understanding of the murder inequalities. We could also go Columbo here with "just one more thing," but I'm a Murder, She Wrote fan.

But what if the murderer used an absolute value inequality? We use them all the time in Cabot Cove.

(Side note: This still is from "One White Rose for Death," Season 3 Episode 4. For those of us born too late and too far from Broadway to see the original production of Sweeney Todd, this episode is a fantastic opportunity to watch Len Cariou and Angela Lansbury work together. And in a later episode his character plays a priest. No meat pies, though.)

Why are you staring at me like that?

Fine, fine, back to the nerdiness you signed up for.

If the absolute value is on the LESS THAN side:

Solve the inequality |2x - 1| + 3< 6

1. Isolate absolute value portion on the left side of inequality. If it's not already there, rewrite it that way (e.g. -- 6 > |2x - 1| + 3 = |2x - 1| + 3 < 6). At least that's what Idiot's Guide says. It may not matter that much.

|2x - 1| + 3 - 3 < 6 - 3

|2x - 1| < 3

2. Drop the absolute value bars and create a compound inequality with the opposite of the "greater than" integer. If you set things up correctly it's the number to the right of the < sign.

-3 < 2x - 1 < 3

3. Solve the compound inequality

-3 + 1 < 2x - 1 + 1 < 3 + 1

-2 < 2x < 4

-1 < x < 2

Even though this example uses <, < uses the same process.

If the absolute value is on the GREATER THAN side:

Here's where it turns out there are two murderers answers. It gets complicated. Stay with me until the trap is sprung.

Solve the inequality |2x + 5| - 4 > -1

1. Isolate the absolute value just as you did with "less than."

|2x + 5| - 4 > -1

|2x + 5| - 4 + 4 > -1 + 4

|2x + 5| > 3

2. Split this into TWO inequality statements. One simply drops the absolute value bars. The other one drops the absolute value bars, flips the sign to "less than," and negates the integer. Put an "or" between them.

2x + 5 > 3     OR     2x + 5 < -3

3. Solve each inequality separately.

2x + 5 - 5 > 3 - 5     OR     2x + 5 - 5 < -3 - 5

2x > -2     OR     2x < -8

x > -1     OR     x < -4

In summation:

|x + a| < b = ONE answer in the form of a compound inequality

|x + a| > b = TWO answers with an OR statement between them

Graphing inequalities

If graphing on a number line open dot means "but not including," filled dot means "and is equal to."

If graphing on a coordinate plane, dashed line means "but not including," solid line means "and is equal to."

Absolute value inequalities will have V-shaped graphs, not lines.

I have no desire to draw examples in Photoshop. Go look at someone else's work. Number line, coordinate plane

Tomorrow I will explain matrices as best I can.

And I'm Stuck

Word problems FTL.

I downloaded worksheets from Kuta Software for more practice material. Systems of equations went well. Mixture problems required some Googling. Once I got a clearly worded example I was set. Everyone has a tendency to do it with things they understand well, but super math minds go all George Lucas when asked to explain their work.

What do you mean you don't see your motivation? It's right there. I wrote it.



When I got to the rate of work problems my decoder skills hung on through "calculate how long two people can do a job together if A works at x rate and B works at y rate. Then came "If A and B work together at x rate, and B works alone at y rate, how long will it take A to work alone?" It should be a simple matter of taking the equation I was already using and solving for a different variable.

And yet it is not proving so simple.

Working together, Paul and Daniel can pick forty bushels of apples in 4.95 hours. Had he done it alone it would have taken Daniel 9 hours. Find how long it would take Paul to do it alone.

I set up my equation

(40 bushels / x hours) + (40 bushels / 9 hours) = (40 bushels / 4.95 hours)

My first thought it to cancel out all those 40s by multplying by 1/40, which gets me to

1/x + 1/9 = 1/4.95

Okay...I still have fractions. Which would be totally okay IF 1/4.95 didn't translate to .202020202.... Time for the homework helpers at algebra.com. (Tip: If you have a child enrolled in algebra who aces their homework but bombs tests, block this site and others like it. Actually just block it until you need to check their homework.)

One solution pulls the number 44.95 out of their Lucas machine, plugs it into the equation, and gets the answer. Where did that number come from?!?!?!?! Another solution explains it by having me multiply across the equation by 9(4.95). Which...I guess that makes sense because I want to get everything to a denominator of one. Why couldn't I see this yesterday? Ugh. The library computer is flashing that my session ends in ten minutes, so I'll skip the work and tell you the answer is 11.

Must post about inequalities before Monday. Must must must. Matrix starts Monday at all costs. Books are due February 8. Worst case scenario I check them out again once they hit the shelf. I doubt anyone wants to fight over algebra books.


Update: After my library session timed out I finally had that epiphany I was searching for. It wasn't multiply everything by 9(4.95). It was cross multiply everything by 9 * 4.95 * x, canceling out as appropriate. So yes, 44.55 came from somewhere, but I couldn't figure out how it went from one x to two.

For my own reference when Sue is having an algebra headdesk moment:

1/14/13 -- I just realized those equal signs in the middle should be multiplication signs. I spent so long trying to remember how to draw lines in Photoshop I forgot what kind of lines to draw! I'll fix them. Someday. Until then use your imagination.

equals 44.55

 equals 4.95x

equals 9x

SO

44.55 + 4.95x = 9x

44.55 = 4.05x

11 = x

Halle-frickin-llujah!

Somethings don't translate well from paperwork to computer notation. And for that there is Photoshop.

Crop tool only works on hands

Word Problems as They Should Be Written

It's that time -- WORD PROBLEMS

Dun, dun, DUN!!!!!

Word problems truly are Idiot's Guide's sole deficiency. The only word problems are in the last chapter and deals more with d = rt as opposed to "If Mary has three apples..." I don't need practice with d = rt! I remember that one! I don't remember divining how many apples Mary has out of no practical information! Do I look like Shawn Spencer to you? "Luckily" Everything is rife with them. I rewrote them with a fresh side of snark.

1. Kristen's grandfather was a college math professor for many years and thinks people enjoy his little challenges. He keeps a jar of coins with only quarters, nickels, and dimes. One day Grandpa tells eleven-year-old Kristen, "There's $48.50 in my silver coin jar. The number of Roosevelts is 20 more than the number of Washingtons, and there are 30 more Jeffersons than Roosevelts. If you can figure out how many of each coin there are by the time your mother gets home, you can have the jar."

Real life answer: Kristen waits until Grandpa takes a nap, takes the money, counts the number of coins, and pays the neighboring unemployed CalTech grad in WoW tokens to come up with the needed equation. Grandpa keeps his word and gives her the jar -- but not the money. Because Grandpa is a jerk like that.

2. Sonya's idiot assistant forgot to mail a very important contract to a client. She also didn't pay the internet bill so email is out of the question. Now Sonya has to take it to the client who is 6 hours away. She gets on the interstate at 7 a.m. and sets her cruise control at 65 mph. Minutes later Allison, aforementioned idiot assistant, realizes she forgot to put the contracts in Sonya's briefcase. She flies out the door with the contracts, jumps in her car, and sets off after Sonya. She gets on the interstate at 7:30 a.m. What is the lowest speed she can set her cruise to catch up with Sonya in less than two hours and hopefully keep her job?

Real life answer: Allison catches up with Sonya but is pulled over by a cop as they're exiting the interstate. Allison gets a $150 ticket. Once Sonya secures the contract in her briefcase she fires Allison. As Sonya drives away she neglects to realize she left the briefcase sitting on the road beside her car. Allison snatches it, throwing it in a drainage ditch on her way back home.

I'm running out of snark steam.


3. Xiang, Amy, and Jiang are 3 years apart in age. Xiao is the youngest and Jiang is the oldest. If 30 more than the sum of Xiao's age and Amy's age is 3 times Jiang's age, can Amy vote in the U.S. general election? 

Real life answer: Amy is old enough to vote in the US general election, but she won't because she thinks it's pointless. Great civic attitude, Amy.

4. Juanita opens a pottery studio in an upscale area of town. Her expenses for total 15n + 320 per day when she has n pieces of pottery to sell. If Juanita charges $28 per piece, a totally realistic price for handmade pottery, how many pieces must she sell each day to make a profit of at least $100 per day?

Real life answer: After the 1000th customer who (1) claims they could totally make that pot for a third of the price, (2) tries to haggle her down to Walmart prices, and (3) allows their children to break pieces and refuses to pay for the damage, Juanita stages a kiln accident that consumes her entire studio for the insurance money.



Actual Mathematical Answers

1.) Solve for quarters. BUT any expression of x must be multiplied by the value of the coin the expression represents. And despite the book examples don't use q, d, or n. I couldn't get past the coin value using variables representative of the coin names.

#quarters = x
#dimes = 2x + 20 (20 more than 2 times quarters)
#nickels = 2x + 50 (30 more nickels than dimes)

25cents(x) + 10cents(2x + 20) + 5cents (2x + 50) = 4850cents
25x + 20x + 200 + 10x +250 = 4850
55x = 4400
x = 80 quarters (or $20.00)

Now solve the other expressions for x

2(80) + 20 = 180 dimes ($18.00)
2(80) + 50 = 210 nickels ($10.50)



2.) I actually handled this one mostly by myself. Probably because I see it as a physics rather than an algebra problem, and while I don't take road trips I drive a lot. I have to figure out how far Sonya travels in two hours AND how far she travels in her 30 minute head start (expressed as .5 hours) to account for it in Allison's catch up speed.
d = (65)(2)
d = 130mi in 2 hours

d = 65(.5)
d = 32.5mi in .5 hours

The formula for Allison's catch up is what I had to double check. I wasn't completely sure where that head start factored in.

130 =  2r - 32.5
162.5 = 2r
81.25 = r

The book gives this as the accepted answer. I consider the actual answer 82 since (a) Allison needs to catch up in less than 2 hours and 81.25 will take exactly 2 hours (perhaps this is best expressed as 81.25 < r) and (b) how are you going to set your cruise somewhere between 81 and 82 miles?



3. Everything is nice enough to prompt me to solve for Amy's age (a), which makes sense because she's the middle of the group. I know Xiao is three years younger (a - 3) and Jiang is three years older (a + 3). Now what to do with that whole 3 times older than yada yada yada and how to set it up.

[Amy's age (a) + Xiao's age (a - 3)] + 30 = 3(a + 3) Jiang's age

a + (a - 3) + 30 = 3(a + 3)
2a - 3 + 30 = 3a + 9
2a + 27 = 3a + 9
-a = -18
a = 18



4. I totally blew this the first try.

15n + 320 ≤ 28n + 100
15n ≤ 28n - 220
-13n ≤ -220
13n ≤ 220
n ≤ 16.92

Even at $28 a pop that doesn't feel right. *look up answer* *go back in chapter for example* The $100 should've gone with the expense formula, not the retail price. And I don't need the ≥ sign, only the >? But it says at least which usually translates to "equals to"...fine. Whateves.

28n > 15n + 320 + 100
28n > 15n + 420
13n > 420
n > 32.30 or 33 pieces per day



I have to google more problems to work. Only the physics-ish one came naturally.

No Wonder I Was Making Great Progress

I've been saying for days I was about to start the matrix, right? Monday night after I posted my blog, ate some cake, and pondered leaving my five year old on someone's doorstep (it was a bad Mama day) I cracked open Everything to practice what I've done so far. My planned quick practice turned into three hours of BWUH? because Everything just had to be written by someone just like my Algebra I teacher. I didn't close the book until 3:30 a.m.

Idiot's Guide is amazing at breaking things down, but so far it hasn't thrown the same type of trick problems at me. The things my teacher would've said, "I just taught you how to do this with an example -- apply what I just said to something that looks absolutely nothing like what we just did!" Fractions with equations in both the numerator and the denominator. Equations with mixed numerals. The dreaded word problems.

I knew it was all going too smoothly. I'll give myself this much -- I must understand what I'm doing now. I was able to figure out which concepts applied to the equations.


(5x - 4) / 7 = (2x + 9) / 3

I saw this and thought, "What the !@#$ am I supposed to do with this shenanigans? I guess I could make the denominators match."

[(3)(5x - 4) / (3)(7)] = [(7)(2x + 9) / (7)(3)]

(15x - 12) / 21 = (14x + 63) / 21

Multiply by the reciprocal to get rid of the denominators... while writing this I realized I could've just cross-multiplied the whole thing from jump street. This is what I get for doing algebra at 1:30 a.m. I can't believe I did all that extra work. And I did it for three different problems!!!!

15x - 12 = 14x + 63
x = 75



When I got to (2n + 9) / (5n - 2) = 1/3 I freaked out a little. Bearings were quickly restored via the long method. Let's try it the short way with cross-multiplication!

3(2n + 9) = 1(5n - 2)
6n + 27 = 5n - 2
n = -29

No algebra after midnight for me. I added three more steps to this.


The inequalities were really disappointing in Everything. No "less than/greater than or equal to" dilemmas at all. I flipped through and didn't see a section on them at all. Everything Guide to Algebra my rear. There was one answer that threw me. I won't bother with the whole process because I did get it right. The book gave the answer as -6 > x > -18. Idiot's Guide calls this bad math grammar and would have me write it -18 < x < -6. I like that better because it's ordered as the numbers appear on a number line. Is one considered technically correct? Technically correct is always the best kind of correct.

The matrix keeps getting shoved back, so I'm officially setting a new start date of 01/14/13 so I can get my posts about fighting with word problems and inequalities up and not do so much of this back and forth between concepts. Books have to be renewed by the 17th, so I have to finish these books three weeks after that. This will get interesting.