Explain the definitions of there, their, and they're and list examples of each.
"There" is an adverb explaining the place of position of a certain noun." Look at those dogs over there!" In this case we are describing that the dogs are somewhere else than where you are, maybe across the street. "Their" explains possession. "Look, it's Ben and Melissa, their dogs are so cute!" In this case, we are stating the Ben and Melissa are in possession of cute dogs. They're is a contraction of they are. "Look at those dogs, they're so cute!" You use they're in any case you could say "they are." For example, "Look at those dogs, they are so cute."
(1/8) / (5/6)
In order to understand this explanation, please write the fractions vertically. I feel it's important to show that I understand fractions, since fractions are commonly misunderstood. This is meant to read as the fraction 1/8 divided by the fraction 5/6. I find that when explaining the division of fractions, many people do not understand why we flip the second fraction (aka reciprocal) and multiply. While I cannot explain why that works exactly, I remember a teacher showing me/explaining a reciprocal as more of a short cut and it made it much easier for me to understand. So, let's do it the long way first. In order to do that we need to find a common denominator between 8 and 6. This can sometimes take a very long time, but in this case it is 24. So what time 8 equals 24?...3! So you must multiply the top by 3 as well giving you a new fraction of 3/24. What times 6 gives you 24?...4! So you must multiply 4 times 5 as well. Now the second fraction is 20/24. Next we would simply divide... (3/24) / (20/24) the bottoms cancel out because what is 24/24?...1! leaving you with the fraction 3/20. This is the final answer, BUT if you were to multiply 1/8 by the reciprocal of 5/6, you find the answer much faster. To find the reciprocal of 5/6 all you do is flip it! so now the equation looks like this (1/8) X (6/5). When you flip that second fraction, you need to remember to flip the sign as well. Now since 8 and 6 each have a common divisor of 2, you reduce those leaving you with (1/4) X (3/5). Now all you do it multiply across. 1x3=?...3. 4x5=?...20. Final answer? 3/20! Ta-Da!
3(2x+1) -4x= -7
The goal of this equation is to separate the x (or in this case xs) from the rest of the numbers by having x on one side of the equals sign and the numbers combined on the other. The first thing I learned in algebra was Please Excuse My Dear Aunt Sally. The first letter of each word represents the order in which you carry out an equation. Please (parenthesis), Excuse (exponents), My (multiplication), Dear (division), Aunt (addition), Sally (subtraction). ALWAYS take care of parenthesis first. In order to do that, you must multiply the 3 times everything inside the parenthesis. So, 3 times 2x equals what? you only worry about the numbers for this, so 3 times 2 is?...6, so the answer is 6x. 3 times 1 equals?... 3. By doing this, it then changes the equation to 6x+3-4x=-7. Now, since we obviously can't add 3 to 6x if we do not know what x equals, we have to combine like numbers. In this case 6x and -4x are alike and 3 and -7 are alike. (Keep the sign in front of the numbers WITH that number. Let's combine 6x and -4x. What is 6-4?...2! So, just keep that x with it and you get 2x. Now the equation should look like this 2x+3=-7. Let's combine 3 and -7. Here's where it can get a little confusing. This is where we start moving the numbers from one side of the equals sign to the other. In order to do that, you have to cancel out the 3 on the left side. What is the opposite of positive 3?...-3! So carry that -3 over to the other side and combine -7 and -3. What is -7-3?...-10! Now the equation looks like this 2x=-10. ALMOST DONE! Just one more step. Now all we need to do is separate that 2 from the x and move it to the other side. 2x stands for multiplication, so what is the opposite of multiplication?...Division of course. So if we are dividing 2 from 2x, we need to divide the other side by 2. What is -10 divided by 2?...-5! Awesome! So we figured out that x=-5. Just remember to pay attention to your negative signs.