# Equations and Inequalities

### Topics

There are three steps to solving a math problem.

- Figure out what the problem is asking.
- Solve the problem.
- Check the answer.

The step of "Solving the Problem" 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 do. 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 are 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 ft^{2}, what is the area of the circle? Don't let the circle distract you with any of its Jedi mind tricks.

We will 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 are doing—don't get hung up on things like solving particular equations for the time being—and you will see that most of what we are 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 ft^{2} 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 are 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.

- Find hypotenuse of triangle / diameter of circle.
- Find radius of circle.
- Find area of circle.

1. Find hypotenuse of triangle / diameter of 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

We get rid of fractions to find that

9 = x^{2}

and take square roots to find that

*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 radius of 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 area of 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* = π *r*^{2}.

This simplifies a bit to

but that is as nice as it gets. Remember to include units, and therefore the final answer is

- Check the answer.

Does the answer we got make sense? Think about this logically for a second. Since π is about 3, we are 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 is 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." (Still not a lawyer, so let's just move on.)

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 radius of circle" and "Find area of 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 will 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.