Gator Math

Solving an Inequality
"An inequality is similar to an equation. There are two expressions separated by a symbol that indicates how one expression is related to the other. In an equation such as 7x = 49, the = sign indicates that the expressions are equivalent. In an inequality, such as 7x > 49, the > sign indicates that the left side is larger than the right side.

To solve the inequality 7x > 49, we follow the same rules that we did for equations. In this case, divide both sides by 7 so that x > 7. This means that x is a value and it is always larger than 7, and never equal to or less than 7.

The "less than" symbol (<) may also be seen in inequalities. " To remember this, we have the "Alligator rule". The alligator wants to eat the bigger number, therefore his mouth is open to the 10 and not the 5: 5 10


If the number on the left were smaller, the alligator head would face the other way, as in this picture.




3 2

Another symbol is the "not equal" sign , when you see this symbol you know that either it is greater than (>) or less than (<) but it is not equal. For instance if you were to have a b, so it would have to be either a (<) b or a (>) b.

Square Roots

To understand Square Roots, Let us look at some patterns

To practice, some square roots, lets take a look at this square roots wristwatch. On this watch, instead of using the numbers 1,2,3,4,5,6,7,8,9,10,11,12: we use sqaure roots to indicate these numbers. By doing this we can see that

You may see a pattern. The number on the left side of the = sign, mulitplied by itself, gives us the number under the square root sign.

1*1=1, 2*2=4, 3*3=9 4*4=16 5*5=25 6*6=36 7*7=49 8*8=64 9*9=81 10*10=100 11*11=121 12*12=144. That is what it means to have a square root of something.

A way to memorize this is by considering another pattern. If we look at those numbers again, we can see that

When going from the square root of 1 to the square root of 4, we just add 3, after that, we just add the next odd number in the sequence. Here are some more examples:

This diagram I made shows this pattern further.

The Power of 2!!

Exponents indicate the number of times the number in front, is multiplied. As you can see, 2^2 means that the number 2 is multiplied 2 times, which yields 4.

If we want to find what 2^3 is, we would simply take 2^2 and multiply it by 2. In other words we would be saying 2*2*2, which is 2^3, and that yields 8.

By multiplying 2, to each product, we can solve all the way up to 2^10, as indicated.

But how do we know what 2^0 and 2^1 are?

As you can see, we have multiplied each power of 2, by 2 to get the next power in the sequence. Dividing, is undoing a multiplication.

2^1:

So, if we know that: we can say that therefore, 2^1=2. We could also solve this by saying that 2 times itself just 1 time, is the same thing as 2*1, which is 2. Another way to look at it is to say that the exponent indicates how many times the number is being persented. Since the exponent is 1, that means that there is only 1 of the number 2.

2^0:

To solve 2^0, we can do the same thing. we can take 2^1 and divide backwards.

2 never does present itself in 2^0, and therefore 2 is shown 0 times. These last two strategies are not as helpful as the dividing backwards method.

Fractions and their Functions

Fractions: A rational number expressed in the form (in-line notation) or (traditional "display" notation), where is called the numerator and is called the denominator. When written in-line, the slash "/" between numerator and denominator is called a solidus.

Adding fractions

When adding fractions, the denominator has to be the same.

In this problem, because the "b" and "d" were not the same, to come up with a common denominator, they had to be mulitiplied together, b(d)=bd. Whatever function is done to the denominator, has to be done to the numerator as well; this is why the "a" is multiplied by "d" and the "c" is mulitplied by "b".

Subtracting fractions

Subtracting fractions is much like adding them in that the denominators must be the same, so which ever function is done to the bottom, must also be done to the top. For this problem:

The common denominator is bd. Because the "d" is mulitplied by the denominator "b", it must also be mulitplied to the numerator "a", which then gives us the numerator "a(d)" or "ad". When the denominator "d" is mulitplied by the denominator "b", it is also mulitplied to teh numerator "c", which gives us "c(b)" or "cb". When put all together, it yields: ad-bc/bd.

Mulitplying fractions

To multiply fractions, we first multiply the top and then we multiply the bottom, straight across. In this way, it is slightly different than when adding or subtracting fractions. As shown in this problem:

Dividing fractions

When dividing fractions, we cross mulitply which means, the 1st numerator is multiplied by the second denominator. When this happens, the product becomes the new numerator. Next, the second numerator is multiplied by the first denominator, and this new product becomes the new denominator.

In this problem, the "a" is multiplied by the "d" an dthe "c" is mulitplied to the b, yielding a(d)/b(c).

Manuel's Apples

If everyday Manuel brought home a green apple, and every three days he brought home a red one. After three weeks how many red apples will Manuel have brought home? How many green apples will he have brought home?

This problem can be solved using what we call "Algebra". "Algebra is a branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set."

To better understand this concept, let us explore the previous story problem.

Step 1: To solve this problem, we could say that because Manuel gets 1 apple for every day we must figure out how many apples he has after 3 weeks. There are 7 days in one week, so for three weeks we could use the equation 7days=1week, so for 3 weeks, it would be 7 times 3. This can be written algebraically as: 7days=1week, ____days=3w, so 7*3=21, therefore 21 days=3weeks.

Step 2: Because he gets 1 green apple each day, 1green apple=1day, therefore 21greenapples=21days.

21days=

Step 3: Manuel gets one red apple every three days. Over the 21 day period, he will get one red apple every three days. This can be written algebraically as: 21days=3r. We use the variable "r" to represent the red apples.

21 days=3r or 21days=


Step 4: By using this equation, we can divide: 21days/3r=7. Therefore, after 21 days, Manuel will have 7 red apples.

21 days=

Now we can safely say that after 3 weeks, Manuel will have 21 green apples and 7 red apples.

3 weeks=