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Back on the page "Peel the Onion" we listed 2 rules |
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| for peeling stuff away from the X. | ||
| First peel away any term on the side with the X | ||
| that doesn't have an X in it. | ||
| If there's only one term on the side with the X, | ||
| stuff on the other side of the biggest fraction line | ||
| is the stuff that's farthest away from the X. | ||
| Since that page, we have done a lot more shenanigans with X's. | ||
| To keep our Onion Peeling rules up to date | ||
| we need to add another rule ... | ||
| If there is only one term on the side with the X | ||
| and the X is somewhere inside the parenthesis, | ||
| deal with the stuff on the outside of the parenthesis first. | ||
| Now we have 3 rules. | ||
| That's all the new rules, for now anyway. | ||
| The new rule goes between the other two. | ||
| Think of these rules like step by step instructions. | ||
| Rule 1: First peel away any term on the side with the X | ||
| that doesn't have an X in it. | ||
| Rule 2: If there is only one term on the side with the X, | ||
| and the X is somewhere inside the parenthesis, | ||
| deal with the stuff on the outside of the parenthesis first. | ||
| Rule 3: If there is only one term on the side with the X, | ||
| stuff on the other side of the biggest fraction line | ||
| is farthest away from the X. | ||
| We have used parenthesis before | ||
| but there is one special parenthesis that we need to talk about. | ||
| It wears a disguise. | ||
| The disguise looks like this: | ||
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| Enough already with all this silly theory. | ||
| Let's do some problems !!! | ||
| Example: | ||
| Solve for X ... | ||
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| We start with rule 1. | ||
| It says that any term that has no X in it | ||
| on the same side of the equation as the X | ||
| is farthest away. Peel it away first. | ||
| That means we start with the "+ 2". | ||
| To get rid of a "+ 2" we subtract 2. | ||
| Anything we do to one side, | ||
| we need to do to the other side too. | ||
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| Now we have one term on the side with the X. | ||
| The X is somewhere inside of the parenthesis | ||
| so we deal with the stuff on the outside | ||
| of the parenthesis first. | ||
| We can get rid of an exponent of 2 | ||
| by taking the square root. | ||
| Remember that we need to do the same thing | ||
| to both sides of the equation ... | ||
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| Now we have two terms on the side with the X again. | ||
| So we go back to rule 1. | ||
| We need to peel away a "- 4" | ||
| so we add 4 to each side ... | ||
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| Example: | ||
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| First we peel away the term with no X in it ... | ||
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| Now we have one term with that sneaky parenthesis | ||
| around everything on the side with the X. | ||
| There is nothing outside of the sneaky parenthesis, | ||
| so we deal with the sneaky parenthesis itself. | ||
| The way to get rid of a square root, | ||
| is to square it. | ||
| Don't forget we need to do that | ||
| to both sides of the equation ... | ||
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| Now we have one term on the side with the X. | ||
| There are no parenthesis. | ||
| We go to rule 3. | ||
| The thing on the far side of the fraction line | ||
| from the X is a 4. | ||
| A 4 in the denominator is like 1/4. | ||
| To peel it away, we multiply by 4/1 ... | ||
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| Now we are back to rule 1. | ||
| Peel the "+ 3" away from the X. | ||
| Subtract 3 from each side ... | ||
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| OK. Let's do one more ... | ||
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Example: |
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| Yuk! What a mess! | ||
| Just take it one step at a time | ||
| and everything will work out. | ||
| First peel away the term with no X. | ||
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| OK, now we have 1 term with parenthesis. | ||
| But what's the furthest outside? | ||
| The 8 or the squared (the 2)? | ||
| Here's the deal. | ||
| The 2 is an exponent. | ||
| An exponent is an instruction that says: | ||
| "multiply the thing on my left times itself | ||
| this many times." | ||
| That means the problem can be written like this: | ||
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| The 8 is the thing that is not part of the X. | ||
| That means it goes away first. | ||
| The 8 is multiplied times the other stuff | ||
| so we get rid of it by dividing ... | ||
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| This looks a lot like the last example now. | ||
| Since we have parenthesis, we need rule 2. | ||
| We get rid of an exponent of 2, | ||
| by taking the square root of each side ... | ||
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| Now we have one term with no parenthesis. | ||
| We do have a denominator, so it's time for rule 3. | ||
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| Now we have 2 terms again. | ||
| That means it's time for rule 1. | ||
| We have a "- 1" to peel away, | ||
| so we do it with a "+ 1" on each side ... | ||
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copyright 2005 Bruce Kirkpatrick |
