Study Guide

Equations and Inequalities - Complicated Equations

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Complicated Equations

As Avril Lavigne would ask: Why do you have to go and make things so complicated, algebra?

Good question, Avril. The truth is that we live in a complicated world, so many of the equations we use are also complicated. We can hardly expect to be able to understand a concept as intricate as nuclear fission with equations as simple as x = 3y. If it were that easy, we wouldn't just need to worry about Iran's nuclear capabilities...we'd also need to keep a wary eye on every middle schooler with an algebra textbook. The world would be a scary place.

The good news is that there are ways of taking seemingly complicated equations and breaking them down into simpler, more manageable chunks. Once you've done that, you'll be able to remove the fractions, or the parentheses, or whatever else it is that's sticking in your metaphorical, mathematical craw. You just need to learn the tricks.

Lucky for you, we know those tricks—and we're eager and willing to share them with you. We only ask that you exercise caution when thinking about defusing a gravy grenade anywhere. You may be trying to be cool, but you look like a fool to us. (Ah, the early 2000s, when musical lyrics were indistinguishable from Shakespearean poetry.)

  • Simplifying Equations

    Sometimes parentheses make equations look more complicated than they actually are. (See how much more complicated this sentence seems?) If you simplify the expressions on each side of the equation before solving it, your life will be much easier.

    Sample Problem

    Solve the equation 9(x – 2) + 3 = 5(1 – x).

    First, simplify each side to find an equivalent equation:

    9x – 18 + 3 = 5 – 5x
    9x – 15 = 5 – 5x

    This looks much better. Now we solve this equation as usual:

  • Eliminating Fractions

    Aw...what did they ever do to you?

    When lots of fractions are involved, there's another way to make an equation look simpler before solving it: get rid of the fractions. Sweep them away, pack them in garbage bags and dump then into the bay. But not really, because that's littering. To get rid of the fractions, we pick a useful number and multiply both sides of the equation by that number. The number is useful if multiplying eliminates all fractions.

    Plus, if the number does a good enough job cleaning up the fractions, maybe we'll see how it does with our bedroom.

    Sample Problem

    Solve the equation .

    Way 1: Subtract 2/3 from each side so that , then simplify that right side.

    Way 2: Find the LCD of the fractions—in this case, 6. Multiply the left-hand side of the equation by 6 and the right-hand side of the equation by 6 to get:

    Oy. So many fractions and parentheses, we'd better simplify this sucker. Fortunately, it simplifies to:

    4 + 6x = 1

    Much better. Notice that there are no longer fractions in the equation, not to mention that the parentheses are gone as well, which is a nice bonus. Now subtract 4 from both sides.

    6x = -3

    So  is the solution to the equation. We wound up with a fraction anyway, but it sure was nice being without them at least for a little while.

    Make sure you understand that getting rid of fractions isn't the same thing as "simplification." When we "simplify," we rewrite the expressions on each side of the =  sign to be tidier, but we don't change the value of either expression. When we eliminate fractions, we're multiplying both sides of the expression by the same number and therefore changing the values of both expressions—but in such a way that the scale is still balanced. Each side is much, much heavier. In fact, we should probably put the whole thing on a sturdier table.

  • Keeping Both Solutions

    Equations of the form x2 = (some positive number) have two solutions: the positive and the negative square roots of that positive number. That's okay—there's room here for a second answer. The more, the merrier.

    What? Of course we're not saying that through gritted teeth.

    Sample Problem

    Solve the equation y2 = 9.

    We've got two options here: y can be either 3 or -3 because (-3)2 = 9 and 32 = 9. Since mathematicians like to abbreviate some things (winner of the Understatement of the Month Award), we write these two answers together as:

    y = ±3

    Sample Problem

    Solve the equation x2 + x – 4 = x.

    Here's the work:

    Be Careful:  Sometimes a problem is set up so that a negative answer doesn't make sense. When there are numbers involved, anything goes, but when it's in the form of a word problem, sometimes you can toss the negative solution. If we're looking for the number of chickens Farmer Ben has let flee the coop, there's no sense in giving him a number that includes negative chickens.

    In these cases, only give the positive root as your final answer, but point out that this is what you're doing. This will convince your teacher and your cute study partner that you know what you're talking about. We probably wouldn't mention the chickens, though.

  • When You Get Stuck

    Like a truck. In the mud.

    Sometimes, when working out a math problem, we get stuck. Our tires are spinning like crazy, but we aren't moving forward a hair. Here's a neat trick: figuring out why we're stuck often tells us what we need to do to get un-stuck.

    Sample Problem

    Solve the equation 

    We haven't seen anything like this equation before. We've seen things like it, but not exactly...we've seen other equations, but we...oh, you know what we mean. The is in the denominator of the fraction instead of in the numerator. Because we don't know what to do about that, let's get the x out of the denominator. Quickly, before it causes any trouble. Let's try multiplying both sides of the equation by x.

    1 = 10x

    From here, we know what to do. Divide both sides by 10 to finish up.

    Next time you get stuck on a problem, write a note in the margin saying, "I don't know what to do here because...(fill in the blank)." For example, you might write, "I don't know what to do here because there's a freaky-looking x in the denominator." See if knowing why you're stuck helps you find a way to get un-stuck. It may surprise you how much simply writing down the point of confusion can help you clear away the cloud. If that doesn't work, you can always try a rain dance.

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