Study Guide

Equations and Inequalities - In the Real World

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In the Real World

Believe it or not, there are lots of jobs where you'll need to know algebra—including some exciting ones involving explosives. Especially jobs where you're hired to blow up inequalities.

Check out the US Military site and search for algebra; it appears in quite a few job descriptions. You know, like You + Algebra = America.

Algebra is useful because it provides a convenient way to solve lots of different types of problems. If we can turn a problem into an equation or inequality, then we can solve the problem by solving the equation or inequality. We don't need to think about the original problem until it comes time to actually write down the answer, when we need to make sure we're answering the right question.

It's like the bus that takes you from Point A to Point B, and you only need to wake up five minutes before you get there to make sure you're getting off at the right stop. Point B, in case you've forgotten.

  • Fitting Things in Spaces

    If you aren't the greatest at maximizing closet space, you may have some direct experience being frustrated by this one. It'll often come in handy to know how much room you're working with and how much of a certain object or material can possibly go there.

    Sample Problem

    Shin is building a shelf to put over his desk and hold his books. Each book is  of an inch thick. Okay, they're issues of Vogue, but same difference. The board Shin is using is 2 ft long. How many "books" can Shin fit on the shelf?

    This problem can be turned into an inequality. We're turning it into an inequality rather than an equation because we might have a little space left over on the shelf. Let x be the number of "books" on the shelf. We'll stop putting quotes around "books" now, since it's exhausting. Paying attention to units, we see the shelf is 2 feet = 24 inches long. Taking this information and writing it in symbols, we find that:

    Now that we have this inequality, we can forget about books for the time being and solve the inequality. To do this, we multiply both sides by .

    Then simplify.

    x ≤ 32

    Now we need to think about books again. Sorry. It'll all be over soon. This answer means Shin can fit up to 32 books on his shelf.

    Okay, you can stop thinking about books now. We told you it would fly by.

    Algebra can also be used to figure out how many cars you can fit in a parking lot, how many boxes of cereal you can fit on a shelf, how many airplane runways you can fit on a piece of land, or how many full-sized marshmallows you can fit in your mouth without choking. Don't try this at home.

  • I Like Abstract Stuff; Why Should I Care?

    Mathematicians can find complications everywhere, even in the seemingly simple idea of putting things in order. If you date a mathematician, your sock drawer will never be the same again.

    In algebra, the phrases "less than" and "greater than" have meanings based on the positions of numbers on the number line, because that's how mathematicians have agreed to order the real numbers. Trust us—it's an ordeal and a half getting all of those guys to agree on anything, so appreciate it for what it is.

    This order makes a whole heap of sense. It stands to reason that 200 should be greater than 4 because you have a lot more chocolate with 200 brownies than with 4 brownies. If you don't, then we shun your brownies. Shun them.

    We can order things besides the real numbers. Pairs of numbers can be ordered using the "lexicographic order." Say what? You know, Lexico. That country just south of Lamerica? Okay, it actually means that one pair is greater than another if its first number is greater than the other pair's first number.

    (1, 4) < (5, 2) since 1 < 5.

    If the first numbers in each pair are the same, we compare the second numbers:

    (3, 1) < (3, 2) since the first numbers in each pair are equal and 1 < 2.

    The lexicographic order is also called the "dictionary order" (the word "lexicon" means "dictionary"), because this order is similar to how we alphabetize words: we compare the first letters, then we compare the second letters, and so on. Look at that—your library shelving skills just improved.

    In some orders, you can't necessarily compare every element to every other element. If Alice, Bianca, and Carol are climbing a mountain as shown below (careful, ladies—climb with your knees!), we could say Bianca is "greater" than Alice since Bianca is higher up the mountain. If we only use strict inequalities, we can't compare Bianca and Carol, because neither of them is higher than the other. Even though Bianca clearly has better hair.

    This mountain example is an instance of a partial order, meaning an order with incomparable elements. Here, Bianca and Carol are both killing it equally, and so cannot be compared. Get it together, Alice. This is getting embarrassing.

    An order in which we can compare any two elements is called a total order. 

  • How to Solve a Math Problem

    There are three steps to solving a math problem.

    1. Figure out what the problem is asking.
    2. Solve the problem.
    3. Check the answer.

    The whole "solving the problem" step will become a little more involved as we get deeper into algebra, but that's okay. What's life without a few challenges? How can highs be highs without the lows? Where—wait, where's our big book of platitudes?

    When solving a problem, we want to break the problem into smaller pieces that we already know how to handle. After we work out the little pieces, we put them together like the pieces of a jigsaw puzzle to get a final answer. If you're no good at putting together puzzles, you might have some trouble here, too. Our advice: sign up for a puzzle-putting-together class. Um, yeah, they have those. Sure.

    Sample Problem

    Below is a circle with a right triangle inside it. Each leg of the triangle has a length of x feet, and the hypotenuse of the triangle is the diameter of the circle. If the area of the triangle is  ft2, what's the area of the circle? Don't let the circle distract you with any of its Jedi mind tricks.

    We'll go through this problem in a lot of detail, and it may seem like a bit of a marathon. You might want to grab a snack first; maybe take a power nap. Try to follow the general outline of what we're doing—don't get hung up on things like solving particular equations for the time being—and you'll see that most of what we're doing is breaking the problem down into tiny pieces that we know how to deal with. Ready? Okay. Let's start being destructive!

    • Figure out what the problem is asking.

    This part is clear. We want to find the area of the circle. Our final answer will have units of ft2 since we can see from the problem that all dimensions are measured in feet. There's a certain irony in the fact that we need to square something to arrive at the dimensions of a circle, don't you think? No? Not a fan of exponent/shape humor? Wow, we're really striking out today.

    • Solve the problem.

    This is the involved part. Before diving in, let's devise a general plan of attack. Battles tend to go better if you have a plan of attack. If you run onto the field waving your mace about, bad things can happen. Also, why are you still using a mace? It's the 21st century.

    We want the area of the circle. If we have the radius, we can find the area. We can find the radius of the circle using the triangle because the hypotenuse of the triangle is the diameter of the circle. As you can see, this is an "A leads to B leads to C" type scenario. Like how that old lady swallowed the cow to catch the goat to catch the dog to catch the cat to catch the bird to catch the spider to catch the fly. We don't know why she swallowed the fly. Perhaps she'll die. Wow, this paragraph ended on a morbid note.

    Okay. Here's the plan.

    1. Find the hypotenuse of the triangle / diameter of the circle.
    2. Find the radius of the circle.
    3. Find the area of the circle.

    1. Find the hypotenuse of the triangle / diameter of the circle.

    If we knew x, we could use the Pythagorean Theorem to find the hypotenuse of the triangle. Rather than wallowing in the fact that we don't know it, which, quite honestly, would be less work, let's be proactive and find it. We can find x using the relationship between x and the area of the triangle.

    We know the area A of a triangle is given by the formula where b and h are the base and height of the triangle. Or the "bottom" and "highness" of the triangle...whatever works for you. We know the area of this particular triangle is , and we know both b and h are equal to x, so:

    Now multiply both sides by 2 to drop those gross fractions:

    9 = x2

    And take the square root of both sides:

    x = ±3

    Because x is a length, x can't be negative. You've never heard of a horse winning a race by negative 3 lengths, right? We toss out the answer x = -3 and keep x = 3.

    Almost there. Now we use the Pythagorean Theorem (check your back pocket—it should be there) to find the hypotenuse of the triangle.

    {hypotenuse}2 = (3)2 + (3)2 = 18

    So the hypotenuse is . Again, we toss out the negative square root since it wouldn't make sense here. Because the hypotenuse and the diameter of the circle are the same, the diameter of the circle is . Isn't it ironic that the diameter of a circle is the square root of—oh, wait, we made that joke already.

    2. Find the radius of the circle.

    We hardly need to do anything here. In fact, try doing this part with one hand tied behind your back. You may need a friend to help with the knots. We know the diameter of the circle is , and we divide the diameter by 2 to find the radius. The radius of the circle is . Okay, now wiggle free of your binds and let's move on.

    3. Find the area of the circle.

    This step doesn't need much work either. We substitute the radius of the circle into the formula for the area of a circle: A = πr2.

    This simplifies a bit to:

    But that's as nice as it gets. Slap those units on the end to get our final answer:

    • Check the answer.

    Does the answer we got make sense? Think about this logically for a second. Since π is about 3, we're claiming that the area of the circle is about three times the area of the triangle. We can fit a copy of the triangle in the bottom half of the circle, and there's about a triangle's worth of space left over on the edges of the circle, and therefore, the answer we got is believable. This is what a lawyer would call "corroborating evidence."

    Congratulations! You've survived! A few cuts and scrapes, but nothing that won't clear up with a bit of Neosporin and a few kisses from Mom. Now go back and look over the problem again. Really, go look at it again. We'll wait.

    Notice that a lot of steps, like "Find the radius of the circle" and "Find the area of the circle," didn't take much work or brainpower. Sure, we had to solve one equation, but if you practice solving equations, you won't need to think hard about how to do that either. Hopefully, we'll eventually get to the point where you don't need to use your brain at all.

    For many problems, the hardest part is figuring out what tiny pieces to break the problem into. Once you get that far, the tiny pieces often take care of themselves. Golly. They grow so fast.

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