Study Guide

Algebra Introduction

    Algebra Introduction

  • Intro

    Pop Quiz Time!

    Which of the following will NOT give you a splitting headache?

    (A) Algae
    (B) Algeria
    (C) Algebra
    (D) Algebra textbook

    The answer, believe it or not, is (C) Algebra. Blue-green algae can cause terrible headaches, algebra textbooks hurt as much as solid brick when you slam your head against them, and have you ever looked into booking a flight to Algeria? Headache central.

    Algebra—basic algebra, anyway—isn't that bad. You may have heard some not-so-great things; in fact, your older brother may have described it as a method of torture that sadistic teachers enact upon students for their own twisted pleasure. He’s just messing with you, though. That’s what older brothers do. Besides, he slept through high school. What does he know?

    Honestly, algebra is one of the most straightforward and rational math courses you'll come across...even when we're talking about irrational numbers. It's all about moving from one step to the logical next step, like a brilliant but prickly Sherlock Holmes-type. You can put the Advil away—you won’t be needing it. Keep the magnifying glass and deerstalker hat, though.

    Why Should I Care?

    Isn’t it enough that we asked nicely?

    Oh, all right. You should care because your life depends on it. We don't mean that algebra itself is a matter of life and death, but your quality of life will be much improved if you develop a solid understanding of algebraic principles. Don't look at us like that; we speak the truth.

    Sound like an exaggeration? Think about it. You'll have a leg up in a whole slew of applicable real-life scenarios, like figuring out payments, tips, taxes, and receipts—if you're an algebra whiz, fewer mistakes mean more dollars in your pocket. Algebra can help you make better cupcakes, since it'll let you adjust a recipe to make a half or double batch. It can even show you the perfect trajectory at which to shoot a basketball so that you'll sink that jump shot. Not to mention its massive usefulness in all kinds of different career paths. Whether your heart is set on becoming an engineer, an accountant, a computer guru, or a statistician, algebra will play a big part in what you do on a daily basis. 

    This is, of course, an incomplete list. Clowns also use algebra. Otherwise, how could they possibly figure out how to fit so many of them into a single car? You can't fit twelve clowns into a Prius without some serious mathematical wizardry.

    While algebra can be a bit dry and tedious, we will absolutely, positively not allow it to be that way on Shmoop. We've packed our guides so full of awesomeness that you'll think you've died and gone to the circus. You see? All roads lead back to clowns...or at least have clown cars driving on them.

    Algebra is everywhere and always will be, like it or not. Let's show it who's boss.

    • What is Algebra?

      An easier question to ask might be, "What isn’t algebra?" There's algebra in the money we make, save, and spend; there's algebra in the construction of everything from kitchen sinks to Priuses to skyscrapers; there's algebra in the preparing, baking, and cooking of food; and there's algebra in watching TV. But only if you're attempting to figure out the perfect distance you should be from your television set so that your eyes don’t get all buggy.

      Algebra is a branch of pure mathematics that deals with the rules of operations and solving equations. Basically, if you’ve got a few pieces of information and you're looking for one last bit that'll make everything else come together, algebra is there for you. Like Friends.

      If you think of all of mathematics as a giant wagon wheel with each branch as a separate spoke, algebra would be the hub. It can be directly applied to just about every other type of math you can think of...and probably quite a few that you can’t. To understand any sort of advanced mathematical concept, in other words, first we've got to wrap our heads around the basics: algebra.

      Fortunately, the basics are pretty basic. Imagine that.

      To get your feet wet, here's a quick example video of algebra in action.

    • Basic Elements

      You wanted to learn the basic elements of the basics? Very well. Shmoop is a river to our people.

      Algebra has a lot to do with the fundamental arithmetic operations, otherwise known as  +, -, ×, and ÷. By now, you may have already "put two and two together," as you should have been able to "take away" from this paragraph that algebra focuses on only the very essentials of the mathematical universe. You don’t need a knowledge of quantum mechanics, you’re not going to be asked to write any complex equations that take up half a chalkboard, and you'll be presented with very few unfamiliar numbers or symbols, although we can't promise that you won't see any.

      In algebra, we introduce the idea of a variable, which is just a secret letter that stands for something else that hasn't been identified. For example, if you're introducing your friend to someone you've only met once and whose name you can't remember, you may find yourself saying, "Brian, this is...x." You may get some funny looks, but at least it’s better than trailing off completely. It does show some modicum of effort.

      In most cases, we attempt to fill in all of those x’s and y’s with real values, and we’ll provide you with the tools to do just that in any given situation. Sometimes, when more than one variable is present, the best you can do is to simplify an equation: you might still be left with a variable or two, but at least it won’t be such a mess.

      We’ll take a closer look at the number line and help you look at numbers in a whole new way. It'll almost be as if you're seeing them for the first time. Although to be fair, it probably also has a little bit to do with your new contact prescription.

      We’ll delineate between natural, whole, real, rational, and irrational numbers, so you can pick any one of them out of a lineup. (If you haven’t seen π’s mug shot, you need to; it’s hilarious.) You’ll also become best buds with decimals, fractions, and percents, in addition to learning how to convert one to another and perform arithmetic operations on any of them. You can perform other types of operations on them as well...but only if they’ve signed a consent form.

      See? Nothing too scary here. No derivatives or logarithms or inflection points. Just the nuts and bolts of numbers—and we’re about to hand you the wrench.

    • Applications in Trigonometry

      Here's one video that showcases trigonometry in action.

      Trigonometry is an in-depth study of triangles and functions, and it has many of its roots in algebra. As we mentioned, algebra sets the stage for a boatload of other mathematical branches, but perhaps none more so than trig. They’re pretty inseparable. They’re like Batman and Robin. Peanut butter and jelly. Super glue, chopsticks, and the palm of your hand. What? You try separating them. We had to learn the hard way.

      We won’t get too far into trigonometric concepts with this guide, but it’s important to be aware of as we take you through the massive overlap between the two. We can combine algebra with geometry to find the third side of a triangle when we have the other two, and we can build on this concept with trigonometry to find and graph the ratios of the triangle’s sides. Any complex-looking equation you may come across in trigonometry—like the horrifying ex + iy = ex(cos y + i sin y), for example—makes use of algebra. You can't just memorize this stuff, pass a test or two, and let baseball statistics push it out of your brain.

      Lock away everything you’re about to learn, and keep the key in a safe place. You don’t want to have to hacksaw your way in later.

  • What is Algebra?

    An easier question to ask might be, "What isn’t algebra?" There's algebra in the money we make, save, and spend; there's algebra in the construction of everything from kitchen sinks to Priuses to skyscrapers; there's algebra in the preparing, baking, and cooking of food; and there's algebra in watching TV. But only if you're attempting to figure out the perfect distance you should be from your television set so that your eyes don’t get all buggy.

    Algebra is a branch of pure mathematics that deals with the rules of operations and solving equations. Basically, if you’ve got a few pieces of information and you're looking for one last bit that'll make everything else come together, algebra is there for you. Like Friends.

    If you think of all of mathematics as a giant wagon wheel with each branch as a separate spoke, algebra would be the hub. It can be directly applied to just about every other type of math you can think of...and probably quite a few that you can’t. To understand any sort of advanced mathematical concept, in other words, first we've got to wrap our heads around the basics: algebra.

    Fortunately, the basics are pretty basic. Imagine that.

    To get your feet wet, here's a quick example video of algebra in action.

  • Basic Elements

    You wanted to learn the basic elements of the basics? Very well. Shmoop is a river to our people.

    Algebra has a lot to do with the fundamental arithmetic operations, otherwise known as  +, -, ×, and ÷. By now, you may have already "put two and two together," as you should have been able to "take away" from this paragraph that algebra focuses on only the very essentials of the mathematical universe. You don’t need a knowledge of quantum mechanics, you’re not going to be asked to write any complex equations that take up half a chalkboard, and you'll be presented with very few unfamiliar numbers or symbols, although we can't promise that you won't see any.

    In algebra, we introduce the idea of a variable, which is just a secret letter that stands for something else that hasn't been identified. For example, if you're introducing your friend to someone you've only met once and whose name you can't remember, you may find yourself saying, "Brian, this is...x." You may get some funny looks, but at least it’s better than trailing off completely. It does show some modicum of effort.

    In most cases, we attempt to fill in all of those x’s and y’s with real values, and we’ll provide you with the tools to do just that in any given situation. Sometimes, when more than one variable is present, the best you can do is to simplify an equation: you might still be left with a variable or two, but at least it won’t be such a mess.

    We’ll take a closer look at the number line and help you look at numbers in a whole new way. It'll almost be as if you're seeing them for the first time. Although to be fair, it probably also has a little bit to do with your new contact prescription.

    We’ll delineate between natural, whole, real, rational, and irrational numbers, so you can pick any one of them out of a lineup. (If you haven’t seen π’s mug shot, you need to; it’s hilarious.) You’ll also become best buds with decimals, fractions, and percents, in addition to learning how to convert one to another and perform arithmetic operations on any of them. You can perform other types of operations on them as well...but only if they’ve signed a consent form.

    See? Nothing too scary here. No derivatives or logarithms or inflection points. Just the nuts and bolts of numbers—and we’re about to hand you the wrench.

  • Applications in Trigonometry

    Here's one video that showcases trigonometry in action.

    Trigonometry is an in-depth study of triangles and functions, and it has many of its roots in algebra. As we mentioned, algebra sets the stage for a boatload of other mathematical branches, but perhaps none more so than trig. They’re pretty inseparable. They’re like Batman and Robin. Peanut butter and jelly. Super glue, chopsticks, and the palm of your hand. What? You try separating them. We had to learn the hard way.

    We won’t get too far into trigonometric concepts with this guide, but it’s important to be aware of as we take you through the massive overlap between the two. We can combine algebra with geometry to find the third side of a triangle when we have the other two, and we can build on this concept with trigonometry to find and graph the ratios of the triangle’s sides. Any complex-looking equation you may come across in trigonometry—like the horrifying ex + iy = ex(cos y + i sin y), for example—makes use of algebra. You can't just memorize this stuff, pass a test or two, and let baseball statistics push it out of your brain.

    Lock away everything you’re about to learn, and keep the key in a safe place. You don’t want to have to hacksaw your way in later.

  • Major Algebra Themes


    Unfortunately, algebra doesn't come with a stirring and memorable theme song that plays at the beginning of every equation, although we think there should be. It would certainly make problem-solving feel more epic.

    Instead, we have certain themes that'll carry us forward throughout the course of this guide, and we’re going to give you a short introduction to each of them right here. It’s just a way to break everything down and make the pieces seem more manageable, like looking at the picture on the front of a huge jigsaw puzzle before you actually start fitting the pieces together.

    • Recognizing Different Types of Numbers

      Ever run into someone who is super excited to see you, but you just can’t place him to save your life? You make some polite small talk about the weather and try desperately to act like you're not experiencing the world's biggest brain fart, but somehow you always get to that moment when he realizes that you have no idea what's going on...or who he is. Cue awkward silence. 

      Yeah, that’s not a great feeling.

      Here's the good news: when it comes to numbers, you're starting with a clean slate. Learn to recognize each type of number at a glance, and you’ll never be caught in any uncomfortable situations. Trust us—you don’t want to fail to recognize an irrational number. They can fly off the handle at the slightest provocation.

      • Natural and Whole Numbers. Natural numbers are the counting numbers, or the numbers that we use to count: 1, 2, 3, 4, 5, etc. Remember the Count from Sesame Street? One natural number, ah-ah-ah. Two natural numbers, ah-ah-ah. That guy sure knew his counting numbers.
         
        Whole numbers are the same as natural numbers, but with the addition of zero. It just got sad watching zero hanging his head while sitting on the bench all day long, so whole numbers invited him to get into the game. Good for zero. You get ‘em, tiger.
         
      • Integers. The set of integers consist of all whole numbers, plus all their negative reflections. Zero is an integer, although he’s the only one without a reflection. (Probably a vampire.)
         
      •  
      • Rational Numbers. This is any number that can be written as the quotient of two integers, otherwise known as a fraction. This includes every integer, because every integer can be written as x/1. How about -34,823/4? Looks weird, but it's totally rational.
         
      • Irrational Numbers. As the name suggests, these buggers can be unpleasant to deal with. They can't be expressed as a fraction, but instead are generally represented by a symbol so we have something to call them. For example, π and  are a couple of famous irrational numbers. We’ll try to get you their autographs, but please understand that they’re very busy.
         
      •  
      • Real Numbers. Gather up all your rational and irrational numbers, and you’ve got your set of real numbers. Basically, it’s every number that isn’t in the set of...
         
      • Imaginary Numbers. Yes, algebra allows you to use your imagination. In fact, your 8-foot-tall imaginary clown friend is even allowed to join you in the SAT testing room when you eventually get around to taking the test. He isn’t allowed to whisper any of the answers to you, however. That would be cheating.
    • Expressions

      An expression is just a string of numbers and/or variables, with no equal sign in sight. Part of the reason some people get so freaked out by algebra is that they may see an expression like this:

      3x + y2

      ...and assume that they’re looking at some foreign language. What are numbers and letters doing, jumbled together like some sort of keyboard free-for-all? And what is that little "2" doing floating up there near the top? What is it, filled with helium?

      Once you understand what these numbers and symbols mean, though, everything becomes easier to handle. For example, say you have a vegetable garden with 3 tomato plants. You high roller, you. How many tomatoes are there on each plant? That’s what x stands for. Suppose that you also went through the garden and sprinkled onion seeds in a square plot, with the same number of onions in each row and column. That’s what the y stands for, so there are y × y = y2 onions in that square of dirt.

      If we're told that each tomato plant yields 4 tomatoes, we know that x = 4 and that there are 12 total tomatoes in your garden (3 × 4 = 12). Then we find out that 3 onion seeds were planted in each row and column, so that's 32 = 9 total onions. How many total veggies are in our garden?

      Just plug everything we know into our original expression:

      3x + y2
      3(4) + 32
      12 + 9 = 21 veggies

      Meanwhile, you're sitting there and chowing down on that Big Mac while we're counting all the produce in your garden. How ironic.

      Algebra is simply a way of abbreviating real-life scenarios—think of it as a type of shorthand for dealing with calculations that can get much more complicated than tallying up tomatoes and onions. Unless you're allergic to tomatoes, in which case we extend our condolences. Understanding what all these abbreviations mean will help you out big-time in the long run. 

      Pinky swear.

    • Word Problems

      You might be getting ready to dive into the wonderful world of numbers, but that doesn’t mean you’ll be able to escape from English. Sorry, mathletes. Since algebra is simply a way of abbreviating real-life scenarios, it only makes sense that it'll be inextricably linked with written language.

      That's right. Word problems are so fancy that we're busting out the extra syllables. 

      Let's take a look at an example.

      Your mom gives you $50 to go to the store and buy milk (she didn’t have anything smaller, unfortunately...for her). Suppose you walk out with milk that costs $3.15, a box of cookies that costs $4.50, and a ton of magazines. When you get home, you hand your mom $2.35 in change. We have two questions for you:

      1. How are you going to get out of this one, especially after last week’s inflatable-pool-in-the-house fiasco?
      2. How much did you spend on all those magazines, anyway?

      To solve this, all you have to do is turn the story into an equation. You started with $50, then spent $3.15, $4.50, and an unknown amount. After that, you gave $2.35 back to your mother. We can rewrite the situation this way, using the variable x for the unknown amount:

      $50 – $3.15 – $4.50 – x – $2.35 = 0

      If we combine the amount of money you spent and set that equal to the $50 you were originally given, we get this:

      $50 = $10 + x

      From here, you could subtract 10 from both sides, but it’s pretty easy to see from a glance that you must have spent a whopping $40 on those magazines—and we’re pretty sure you didn't buy any back issues of Good Housekeeping. On the upside, you’ll have plenty of time to peruse them all while you’re grounded for the next three days.

    • Graphs

      Here's a video that shows graphs in action.

      When you have an equation like 2x + 3y = 6 and no other clues, sometimes all you can do is plug in numbers randomly and hope that everything works out.

      No, seriously. That's a real math thing.

      Take that equation, for example. If you plug in a few random numbers for x, you might notice that you get results like these:

      x = 0, y = 2


      To represent this data visually, we draw a graph with an x-axis and a y-axis and plot these points like so:

      Either those dots are starting to form a line, or that’s an awfully big coincidence. (It’s the first one.) If you keep plugging in random numbers, you'll continue to get points that fall along this same line. It's mathmagic. You can therefore represent all the possible solutions for both variables by connecting the dots on your graph.

      As with all algebra, we're not just talking about some arbitrary line and some arbitrary grid. This line can be used to represent real information. Think about that vegetable garden you were tending. (Can't remember? Oh, that's right, we were the ones counting your vegetables.) If your tomatoes seem to be growing at the same rate as your onions, you should be able to predict how many onions grow in your garden for every tomato. This data is important, because you have to know how many toppings you’ll be able to put on top of your burger.

  • Recognizing Different Types of Numbers

    Ever run into someone who is super excited to see you, but you just can’t place him to save your life? You make some polite small talk about the weather and try desperately to act like you're not experiencing the world's biggest brain fart, but somehow you always get to that moment when he realizes that you have no idea what's going on...or who he is. Cue awkward silence. 

    Yeah, that’s not a great feeling.

    Here's the good news: when it comes to numbers, you're starting with a clean slate. Learn to recognize each type of number at a glance, and you’ll never be caught in any uncomfortable situations. Trust us—you don’t want to fail to recognize an irrational number. They can fly off the handle at the slightest provocation.

    • Natural and Whole Numbers. Natural numbers are the counting numbers, or the numbers that we use to count: 1, 2, 3, 4, 5, etc. Remember the Count from Sesame Street? One natural number, ah-ah-ah. Two natural numbers, ah-ah-ah. That guy sure knew his counting numbers.
       
      Whole numbers are the same as natural numbers, but with the addition of zero. It just got sad watching zero hanging his head while sitting on the bench all day long, so whole numbers invited him to get into the game. Good for zero. You get ‘em, tiger.
       
    • Integers. The set of integers consist of all whole numbers, plus all their negative reflections. Zero is an integer, although he’s the only one without a reflection. (Probably a vampire.)
       
    •  
    • Rational Numbers. This is any number that can be written as the quotient of two integers, otherwise known as a fraction. This includes every integer, because every integer can be written as x/1. How about -34,823/4? Looks weird, but it's totally rational.
       
    • Irrational Numbers. As the name suggests, these buggers can be unpleasant to deal with. They can't be expressed as a fraction, but instead are generally represented by a symbol so we have something to call them. For example, π and  are a couple of famous irrational numbers. We’ll try to get you their autographs, but please understand that they’re very busy.
       
    •  
    • Real Numbers. Gather up all your rational and irrational numbers, and you’ve got your set of real numbers. Basically, it’s every number that isn’t in the set of...
       
    • Imaginary Numbers. Yes, algebra allows you to use your imagination. In fact, your 8-foot-tall imaginary clown friend is even allowed to join you in the SAT testing room when you eventually get around to taking the test. He isn’t allowed to whisper any of the answers to you, however. That would be cheating.
  • Expressions

    An expression is just a string of numbers and/or variables, with no equal sign in sight. Part of the reason some people get so freaked out by algebra is that they may see an expression like this:

    3x + y2

    ...and assume that they’re looking at some foreign language. What are numbers and letters doing, jumbled together like some sort of keyboard free-for-all? And what is that little "2" doing floating up there near the top? What is it, filled with helium?

    Once you understand what these numbers and symbols mean, though, everything becomes easier to handle. For example, say you have a vegetable garden with 3 tomato plants. You high roller, you. How many tomatoes are there on each plant? That’s what x stands for. Suppose that you also went through the garden and sprinkled onion seeds in a square plot, with the same number of onions in each row and column. That’s what the y stands for, so there are y × y = y2 onions in that square of dirt.

    If we're told that each tomato plant yields 4 tomatoes, we know that x = 4 and that there are 12 total tomatoes in your garden (3 × 4 = 12). Then we find out that 3 onion seeds were planted in each row and column, so that's 32 = 9 total onions. How many total veggies are in our garden?

    Just plug everything we know into our original expression:

    3x + y2
    3(4) + 32
    12 + 9 = 21 veggies

    Meanwhile, you're sitting there and chowing down on that Big Mac while we're counting all the produce in your garden. How ironic.

    Algebra is simply a way of abbreviating real-life scenarios—think of it as a type of shorthand for dealing with calculations that can get much more complicated than tallying up tomatoes and onions. Unless you're allergic to tomatoes, in which case we extend our condolences. Understanding what all these abbreviations mean will help you out big-time in the long run. 

    Pinky swear.

  • Word Problems

    You might be getting ready to dive into the wonderful world of numbers, but that doesn’t mean you’ll be able to escape from English. Sorry, mathletes. Since algebra is simply a way of abbreviating real-life scenarios, it only makes sense that it'll be inextricably linked with written language.

    That's right. Word problems are so fancy that we're busting out the extra syllables. 

    Let's take a look at an example.

    Your mom gives you $50 to go to the store and buy milk (she didn’t have anything smaller, unfortunately...for her). Suppose you walk out with milk that costs $3.15, a box of cookies that costs $4.50, and a ton of magazines. When you get home, you hand your mom $2.35 in change. We have two questions for you:

    1. How are you going to get out of this one, especially after last week’s inflatable-pool-in-the-house fiasco?
    2. How much did you spend on all those magazines, anyway?

    To solve this, all you have to do is turn the story into an equation. You started with $50, then spent $3.15, $4.50, and an unknown amount. After that, you gave $2.35 back to your mother. We can rewrite the situation this way, using the variable x for the unknown amount:

    $50 – $3.15 – $4.50 – x – $2.35 = 0

    If we combine the amount of money you spent and set that equal to the $50 you were originally given, we get this:

    $50 = $10 + x

    From here, you could subtract 10 from both sides, but it’s pretty easy to see from a glance that you must have spent a whopping $40 on those magazines—and we’re pretty sure you didn't buy any back issues of Good Housekeeping. On the upside, you’ll have plenty of time to peruse them all while you’re grounded for the next three days.

  • Graphs

    Here's a video that shows graphs in action.

    When you have an equation like 2x + 3y = 6 and no other clues, sometimes all you can do is plug in numbers randomly and hope that everything works out.

    No, seriously. That's a real math thing.

    Take that equation, for example. If you plug in a few random numbers for x, you might notice that you get results like these:

    x = 0, y = 2


    To represent this data visually, we draw a graph with an x-axis and a y-axis and plot these points like so:

    Either those dots are starting to form a line, or that’s an awfully big coincidence. (It’s the first one.) If you keep plugging in random numbers, you'll continue to get points that fall along this same line. It's mathmagic. You can therefore represent all the possible solutions for both variables by connecting the dots on your graph.

    As with all algebra, we're not just talking about some arbitrary line and some arbitrary grid. This line can be used to represent real information. Think about that vegetable garden you were tending. (Can't remember? Oh, that's right, we were the ones counting your vegetables.) If your tomatoes seem to be growing at the same rate as your onions, you should be able to predict how many onions grow in your garden for every tomato. This data is important, because you have to know how many toppings you’ll be able to put on top of your burger.

  • Key Skills


    To excel at algebra problems, you’ll need to hone some of your special algebra powers. That’s right: you’re basically a mathematical superhero. The world of crime is your Number Line, and Dr. Irrational is your arch-enemy. 

    Save us, Captain Algebra; you're our only hope. What? That didn't sound distressed enough to you?

    • Negative Numbers

      There’s more to making numbers negative than just putting a little line in front of them: you have to really understand what a negative number is and how to deal with one in unusual circumstances. 

      For example, is -(-(-(-(-5)))) negative or positive? Who knows? (You do.)

      Clearly, the first negative sign—the one directly attached to the 5—makes the number negative. But the next sign reflects the number back across 0 on the number line, making it positive once again. After we make a few more flips and wait until the dust settles, we discover that this number is indeed negative, but you can’t make that assumption just because there are negative signs coming out your ears.

      If we've got a negative variable in the mix, we can move it to the other side of the equation with addition. Say we're trying to solve for x in the equation 0 = 5 – x. To slide x over to the left side, we'd just add x to both sides and get rid of that negative sign. Like this:

      0 = 5 – x
      x
      = 5 – x + x
      x
      = 5

      Most of the examples in this guide are fairly straightforward, but things can get confusing pretty quickly. (Example one: quintuple negatives.) Take your time with negative numbers, and make sure you get what’s going on before you start slinging them around willy-nilly.

    • Inequalities

      Some numbers get no respect. Take 0, for example, who usually gets the short end of the stick. In fact, people treat him like he’s not even there; he's a complete nobody, a nothing. But this isn’t quite what inequalities are about. Besides, 0 is actually much larger than a lot of other numbers. 

      He just doesn’t throw it back in their faces. He’s bigger than that.

      Inequalities are mathematical statements that compare quantities. If a problem asks us to fill in the right inequality symbol between two numbers like this:

      -7 __ 3

      ...we would insert a "<" (less than) symbol in the blank to indicate that the 3 is the larger of the two numbers: -7 < 3. We can remember this in one of two ways: think of the symbol as the mouth of a hungry alligator that would prefer to go after the bigger meal, or an arrow that's pointing and laughing at the smaller number. Either way, the numbers won’t be getting out of this situation entirely unscathed.

      We’ll also encounter inequality problems that involve variables, as well as ones that feature "greater than or equal to" and "less than or equal to" symbols. These are just like the hungry alligator/pointing finger (...put together, that could end badly), except they've got a little line underneath to show the "or equal to" part.

      4x – 5 ≥ 10

      To solve a problem like this one, add 5 to both sides and then divide both by 4 in order to isolate the variable. Because if anyone’s going to get eaten, it might as well be the lone letter. It’s every number for herself around here.

       
    • Graphing

      We mentioned graphs a little earlier, but we’ll do a lot more than mention them in this guide. In fact, we're going to be getting up close and personal with graphs of all sorts of different shapes and sizes. You’ll learn several different methods for plotting graphs, and you’ll be introduced to graphs with functions that have varied and unique shapes. Like this:

      Or this:

      Or even this:

      Okay, maybe not that last one. Relax, we're kidding. Gawrsh.

      The really heavy-duty graphing won't join the party until we get to trigonometry and calculus, but algebra will at least help you get your feet wet. In fact, you might want to bring along a spare pair of dry socks.

    • Distributive Property

      This video shows the distributive property in action on a slightly advanced problem.

      Officially, we use the distributive property to multiply binomials. If you need a quick refresher (or introduction) to binomials, those are expressions that consist of two terms. You know, as in "a + b" or "Mike + Brittney 4EVA." We’ll go with the first example, since we found the second one carved into the side of a maple tree and we’re not actually sure if it’s public domain.

      If you want to multiply a + b and c + d, the distributive property tells you how to do exactly that. All we need to do is multiply a by both letters in the second term, then multiply b by everything in the second term. That sounds kind of abstract, so here's what we mean:

      (a + b)(c + d) =
      a
      (c + d) + b(c + d) =
      ac +ad + bc + bd

      That may not look too helpful right now, but this thing will be a lifesaver once you’ve got a problem with a few numbers to plug in. A cherry-flavored lifesaver.

    • Important Formulas

      Just a quick video to get you acquainted with one important formula: the quadratic equation.

      Some formulas are so important that they’re top government secrets, or they're known only to the cook staff at Mack and Cheesie’s World Famous Chili Joint. But you have complete and total access to some of the most important formulas out there—and we’re going to recommend that you take full advantage of that.

      You’ll be introduced to such must-knows as the quadratic equation, slope-intercept form, and the distance and midpoint formulas. We won't bog you down with details now, since you’re just getting into this algebra stuff, but let's just say that you'll know all of these backwards and forwards by the time you’re through. 

      On second thought, just concentrate on knowing them forwards. You’ve got enough going on in that head of yours.

  • Negative Numbers

    There’s more to making numbers negative than just putting a little line in front of them: you have to really understand what a negative number is and how to deal with one in unusual circumstances. 

    For example, is -(-(-(-(-5)))) negative or positive? Who knows? (You do.)

    Clearly, the first negative sign—the one directly attached to the 5—makes the number negative. But the next sign reflects the number back across 0 on the number line, making it positive once again. After we make a few more flips and wait until the dust settles, we discover that this number is indeed negative, but you can’t make that assumption just because there are negative signs coming out your ears.

    If we've got a negative variable in the mix, we can move it to the other side of the equation with addition. Say we're trying to solve for x in the equation 0 = 5 – x. To slide x over to the left side, we'd just add x to both sides and get rid of that negative sign. Like this:

    0 = 5 – x
    x
    = 5 – x + x
    x
    = 5

    Most of the examples in this guide are fairly straightforward, but things can get confusing pretty quickly. (Example one: quintuple negatives.) Take your time with negative numbers, and make sure you get what’s going on before you start slinging them around willy-nilly.

  • Inequalities

    Some numbers get no respect. Take 0, for example, who usually gets the short end of the stick. In fact, people treat him like he’s not even there; he's a complete nobody, a nothing. But this isn’t quite what inequalities are about. Besides, 0 is actually much larger than a lot of other numbers. 

    He just doesn’t throw it back in their faces. He’s bigger than that.

    Inequalities are mathematical statements that compare quantities. If a problem asks us to fill in the right inequality symbol between two numbers like this:

    -7 __ 3

    ...we would insert a "<" (less than) symbol in the blank to indicate that the 3 is the larger of the two numbers: -7 < 3. We can remember this in one of two ways: think of the symbol as the mouth of a hungry alligator that would prefer to go after the bigger meal, or an arrow that's pointing and laughing at the smaller number. Either way, the numbers won’t be getting out of this situation entirely unscathed.

    We’ll also encounter inequality problems that involve variables, as well as ones that feature "greater than or equal to" and "less than or equal to" symbols. These are just like the hungry alligator/pointing finger (...put together, that could end badly), except they've got a little line underneath to show the "or equal to" part.

    4x – 5 ≥ 10

    To solve a problem like this one, add 5 to both sides and then divide both by 4 in order to isolate the variable. Because if anyone’s going to get eaten, it might as well be the lone letter. It’s every number for herself around here.

     
  • Graphing

    We mentioned graphs a little earlier, but we’ll do a lot more than mention them in this guide. In fact, we're going to be getting up close and personal with graphs of all sorts of different shapes and sizes. You’ll learn several different methods for plotting graphs, and you’ll be introduced to graphs with functions that have varied and unique shapes. Like this:

    Or this:

    Or even this:

    Okay, maybe not that last one. Relax, we're kidding. Gawrsh.

    The really heavy-duty graphing won't join the party until we get to trigonometry and calculus, but algebra will at least help you get your feet wet. In fact, you might want to bring along a spare pair of dry socks.

  • Distributive Property

    This video shows the distributive property in action on a slightly advanced problem.

    Officially, we use the distributive property to multiply binomials. If you need a quick refresher (or introduction) to binomials, those are expressions that consist of two terms. You know, as in "a + b" or "Mike + Brittney 4EVA." We’ll go with the first example, since we found the second one carved into the side of a maple tree and we’re not actually sure if it’s public domain.

    If you want to multiply a + b and c + d, the distributive property tells you how to do exactly that. All we need to do is multiply a by both letters in the second term, then multiply b by everything in the second term. That sounds kind of abstract, so here's what we mean:

    (a + b)(c + d) =
    a
    (c + d) + b(c + d) =
    ac +ad + bc + bd

    That may not look too helpful right now, but this thing will be a lifesaver once you’ve got a problem with a few numbers to plug in. A cherry-flavored lifesaver.

  • Important Formulas

    Just a quick video to get you acquainted with one important formula: the quadratic equation.

    Some formulas are so important that they’re top government secrets, or they're known only to the cook staff at Mack and Cheesie’s World Famous Chili Joint. But you have complete and total access to some of the most important formulas out there—and we’re going to recommend that you take full advantage of that.

    You’ll be introduced to such must-knows as the quadratic equation, slope-intercept form, and the distance and midpoint formulas. We won't bog you down with details now, since you’re just getting into this algebra stuff, but let's just say that you'll know all of these backwards and forwards by the time you’re through. 

    On second thought, just concentrate on knowing them forwards. You’ve got enough going on in that head of yours.