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Unit 1 (September)
< Click here to access the Learning Targets checklist for Unit One >
Solving Equations: Recall the purpose of 'solving an equation' from last year. You are essentially finding the value that will make the equation true. For example, in the equation x + 1 = 6, the number 5 is the solution because it makes the equation true.
Remember, when 'solving' you are 'undoing' the operations that are affecting your variable. You can solve them in any order you want, however it is usually easiest to solve an equation by doing the opposite of PEMDAS. (SADMEP). When you perform simple pemdas, you tend to create fractions that make further solving more difficult. See the example below:
Notice how you still arrive at the same answer. However, following traditional pemdas you tend to create fractions and decimals which can make further solving more difficult.
Solving equations with variables on both sides of the equation: When you solve an equation, the last line of work that shows your solution has x on one side, and a number on the other side (for example, x = 5). So if you think of it, you are basically trying to get your variables to one side (whether it is x, y, a, b, etc...) and your numbers/values to the other side. With that in mind, if you have a variable on both sides you must algebraically move them all to one side. See below:
Solving equations with variables on both sides of the equation: When you solve an equation, the last line of work that shows your solution has x on one side, and a number on the other side (for example, x = 5). So if you think of it, you are basically trying to get your variables to one side (whether it is x, y, a, b, etc...) and your numbers/values to the other side. With that in mind, if you have a variable on both sides you must algebraically move them all to one side. See below:
Solving equations with no or 'all real number' solutions: Recall that when solving an equation you are essentially looking for values that will make the equation true. With that being said, some equations will never have solutions, and some equations will have an infinite number of solutions (any number could work).
- No solution: For example, look at x + 4 = x + 3. What number (x) could have 4 added to it, and still be equal to itself with 3 added to it too???? This does not make sense, let alone "work". There is no such number that could be plugged in here. Therefore, this equation has no solution. If we algebraically solve this equation (by subtracting x from both sides), we notice that the x's cancel out and we are left with 4 = 3. Since this is not true that means no values could be plugged in to make it true. The solution therefore is 'no real numbers'.
- All real numbers / Infinite solutions: For example, look at x + 5 = x + 5. What number (x) could have 5 added to it, and be equal to the same number with 5 added to it? This sounds redundant, and it is!!! The left side is equal to the right side, so of course anything you plug in for x on the left, will equal the right. Algebraically, when we solve this (again, by subtracting x from both sides first) we get 5 =5 as a solution. This is a true statement, so the equation works, and x appears to be 'missing' from the solution. So one way to think of this solution is simply recognizing that x is absent, so it has no affect on the equation itself. Therefore x could be anything (all real numbers), and there are an infinite number of solutions for x.
Other Equations: The video below explains other algebraic concepts you were taught last year (in addition to some of the concepts above)
Number properties: You are already aware of most basic number properties, however, you are not expected to algebraically show or prove them until Algebra. Take note: for the quiz/test/exam you will not be expected to be able to state the properties, rather, I may give you algebraic rules and ask you if they are in fact true or not. For example, T or F, is a + b = b + a ? Or I may ask you to create a rule for something based on what you already know.
There are many properties, these are simply some of the more common ones that you see:
Commutative, Associative, Distributive etc...
~ Commutative property of addition and multiplication: You can add or multiply in any order you want.
a + b = b + a and a x b = b x a
~ Associative property of addition and multiplication: If given groups of numbers, you can add or multiply groupings in any order you want.
( a + b ) + c = a + (b + c) and (a x b) x c = a x (b x c)
~ Distributive property of multiplication and division: when multiplying a sum or difference by a number, you distribute that number to each part of the sum or difference. Likewise, when dividing a sum or difference by a number, you distribute that number to each part of the sum or difference.
*note: take note of where the multiplying and dividing numbers appear!!! You cannot distribute forwards for division!!! c / (a + b) does not equal c/a+c/b
a (b + c) = ab + ac and (a + b) / c = a/c + b/c
~ Identity property of addition, subtraction, multiplication, and division: If given a number, an operation of what value would return that same number?
a + 0 = a and a - 0 = a and a x 1 = a and a / 1 = a
~ Zero property of addition, subtraction, multiplication: If given a number, an operation of what value would return a value of zero?
*note: You cannot divide a number by 0 to = 0. However, you can divide 0 by a number to = 0. This is a unique zero property.
a + (-a) = 0 and a - a = 0 and a x 0 = 0
Dimensional Analysis (supplemental unit taught if time): Dimensional analysis is a fancy way of saying unit conversions (like converting miles into feet). It is not really a 'new' topic, but it forces you to show your work. For example, if I asked you to convert 3 days into seconds you could probably do it in 30 seconds with nothing but a calculator. Dimensional analysis provides you with a way to prove your calculator work is correct, and allows you to keep track of each step (because not every conversion is as simple as changing seconds to days).
Dimensional Analysis
Dimensional analysis 2
Dimensional analysis example video (check this vid out for example problems with solutions worked out)
Arithmetic Sequences: Arithmetic essentially means 'simple addition' and a sequences is a 'pattern'. Therefore, an arithmetic sequence is a pattern of numbers (or variables, or expressions) where there is a common number being added from one term to the next. For example, 2, 5, 8, 11,.....
There are many properties, these are simply some of the more common ones that you see:
Commutative, Associative, Distributive etc...
~ Commutative property of addition and multiplication: You can add or multiply in any order you want.
a + b = b + a and a x b = b x a
~ Associative property of addition and multiplication: If given groups of numbers, you can add or multiply groupings in any order you want.
( a + b ) + c = a + (b + c) and (a x b) x c = a x (b x c)
~ Distributive property of multiplication and division: when multiplying a sum or difference by a number, you distribute that number to each part of the sum or difference. Likewise, when dividing a sum or difference by a number, you distribute that number to each part of the sum or difference.
*note: take note of where the multiplying and dividing numbers appear!!! You cannot distribute forwards for division!!! c / (a + b) does not equal c/a+c/b
a (b + c) = ab + ac and (a + b) / c = a/c + b/c
~ Identity property of addition, subtraction, multiplication, and division: If given a number, an operation of what value would return that same number?
a + 0 = a and a - 0 = a and a x 1 = a and a / 1 = a
~ Zero property of addition, subtraction, multiplication: If given a number, an operation of what value would return a value of zero?
*note: You cannot divide a number by 0 to = 0. However, you can divide 0 by a number to = 0. This is a unique zero property.
a + (-a) = 0 and a - a = 0 and a x 0 = 0
Dimensional Analysis (supplemental unit taught if time): Dimensional analysis is a fancy way of saying unit conversions (like converting miles into feet). It is not really a 'new' topic, but it forces you to show your work. For example, if I asked you to convert 3 days into seconds you could probably do it in 30 seconds with nothing but a calculator. Dimensional analysis provides you with a way to prove your calculator work is correct, and allows you to keep track of each step (because not every conversion is as simple as changing seconds to days).
Dimensional Analysis
Dimensional analysis 2
Dimensional analysis example video (check this vid out for example problems with solutions worked out)
Arithmetic Sequences: Arithmetic essentially means 'simple addition' and a sequences is a 'pattern'. Therefore, an arithmetic sequence is a pattern of numbers (or variables, or expressions) where there is a common number being added from one term to the next. For example, 2, 5, 8, 11,.....
The recursive method for interpreting arithmetic sequences is best used when you are simply trying to explain a pattern or attempting to find the next number or next few numbers. The recursive method is generally done mentally (for example, what two terms come next in the sequence 4, 7 ,10, ___, ____). However, as far as Algebra is concerned, you will still be expected to use the formula to "prove" that 13 and 16 are the next two terms.
See the examples below. I went ahead and colored coded the formula and the variables so you could see where I was getting my values:
See the examples below. I went ahead and colored coded the formula and the variables so you could see where I was getting my values:
Chapter 2 (September)
Solving Inequalities: You solve inequalities in the same way that you solve an equation. The equal, greater than, less than, greater or equal, and less than or equal are ALL signs that show an equality relationship between two expressions. Therefore, the methods for solving them are generally the same. Except for one thing.... (see below)
Think about what the negative sign means. It also means the 'opposite' of a number. So the opposite of -7 is 7 right? Yes, but -x can be anything more than 7 too (like 8, 9, 10 etc...). Therefore, x (which is the opposite of -x) can not only be -7, but also -8, -9, -10 etc...... Therefore, the inequality sign must change direction.
The simple rule? When you multiply or divide an inequality by a negative number, you must also flip the sign. Period.
Website that explains just about every situation you may encounter when solving a basic inequality
Compound Inequalities: These are inequalities that have more than one boundary. For example, while x > 3 means that x can be any value greater than 3, I can also say (and graph) x > 3 or x < 1. So x can be any value more than 3 OR less than 1. I can also graph x< 3 and x >1. So x can be any value more than 1 AND less than 3. The caveat to the second example is that since any number you choose must satisfy BOTH inequalities (like the #2 for example), you may write this as one big inequality; 1 < x < 3. This is relatively easy to remember, 1 is your lower boundary, 3 is your upper, and x is in the middle of them. Just make sure your signs are correct!!!
Video on solving compound inequalities
How to solve compound inequalities
Website that explains just about every situation you may encounter when solving a basic inequality
Compound Inequalities: These are inequalities that have more than one boundary. For example, while x > 3 means that x can be any value greater than 3, I can also say (and graph) x > 3 or x < 1. So x can be any value more than 3 OR less than 1. I can also graph x< 3 and x >1. So x can be any value more than 1 AND less than 3. The caveat to the second example is that since any number you choose must satisfy BOTH inequalities (like the #2 for example), you may write this as one big inequality; 1 < x < 3. This is relatively easy to remember, 1 is your lower boundary, 3 is your upper, and x is in the middle of them. Just make sure your signs are correct!!!
Video on solving compound inequalities
How to solve compound inequalities
Absolute Value Equations: Recall that absolute value is simply the distance a value is from 0. For example, |x|=6 means what values for x have an absolute value of 6? Well, 6 and -6 would be possible solutions. When given larger equation, like |2x+4|=10, you are being asked the same thing. For what values of x will 2x+4 be 10 away from 0? You know that 2x+4 therefore can be at 10 and -10 since these values are 10 away from 0, so you are essentially solving 2x+4=10 and 2x+4=-10. In a nut shell, when given AVE's (or inequalities...later) you are creating 2 equations to solve and find values for.
Basic solving absolute value equations
Basic solving absolute value equations
Extraneous solutions: An extraneous solution is a solution that was found algebraically (assuming your math was correct), but it does not work in the original equation when you attempt to check it.
Extraneous solutions: An extraneous solution is a solution that was found algebraically (assuming your math was correct), but it does not work in the original equation when you attempt to check it.
Absolute Value Inequalities: Solve them similar to absolute value equations, except keep in mind that the inequality sign changes when you 'turn' one side negative. AVI's are used most frequently in manufacturing where there are acceptable ranges for products. For example, examine the coin photo at right:
The coin on the top clearly has too small of a ridge (yes I know it looks "ok" to you and me, but to a machine, it is not acceptable). The acceptable range might be an average ridge width of .2 centimeters with a range of +/- .03. The computer that examines each coin is programmed using an AVI set to the formula |x - m|= n' where m is the middle value (.02) and n is the acceptable numerical difference (.03), and the equality symbol matches the scenario (<><>=). The machine will accept coins with an x value that satisfies this inequality, but reject coins that do not. The formula works because you are accepting coins that are within .03 cm from the expected width, you do not care if they are +.03 or -.03, so long as the distance is .03 or less. Therefore, you want the absolute value because AV represents a distance from the middle regardless of being higher or lower.
Video on solving absolute value inequalities
Below are a few examples of solving absolute value inequalities. Take note of each unique situation (normal AVI, extraneous solutions, AV on both sides).
Word Problems with Absolute value equations and inequalities: Word problems with absolute value center around your knowledge of applying the concept of "distance". Recall that the equation | x | = 5 is asking for you to find values that are 5 away from 0 (meaning x=5 and x=-5 are solutions). In class, we experimented with the numberline, for example, how would these solutions look and where is their "line of symmetry". We then graphed the solutions to the equations | x - 2| = 5 and | x + 3 | = 5. See below:
As we discussed in class, the first graph is showing us the values that are 4 units away from 0 (our definition of Absolute Value.
However, in the second graph we notice that the solution of 4 is actually the distance that each solution lies from 2 (6 and -2 are both 4 away from 2).
After experimenting with a few more graphs, we can construct the formula below:
*note.....this is not a "real formula". Meaning you cannot 'google' the x-men formula. This is simply a formula we created in class to help us navigate absolute value word problems.
However, in the second graph we notice that the solution of 4 is actually the distance that each solution lies from 2 (6 and -2 are both 4 away from 2).
After experimenting with a few more graphs, we can construct the formula below:
*note.....this is not a "real formula". Meaning you cannot 'google' the x-men formula. This is simply a formula we created in class to help us navigate absolute value word problems.
This formula can be applied to any problems where you can distinguish a central value (the middle or mean), and each end point is equidistant from this middle. For example: