Algebra I
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CA Algebra I: Number Properties and Absolute Value
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Simple Logical Arguments
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CA Algebra I: Graphing Inequalities
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CA Algebra I: Slope and Y-intercept
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CA Algebra I: Systems of Inequalities
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Factoring Quadratics
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CA Algebra I: Completing the Square
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CA Algebra I: Quadratic Equation
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CA Algebra I: Quadratic Roots
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CA Algebra I: Rational Expressions 1
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CA Algebra I: Rational Expressions 2
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CA Algebra I: Word Problems
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CA Algebra I: More Word Problems
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CA Algebra I: Functions
CA Algebra I: Simple Logical Arguments 14-20, simple logical reasoning
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- We're on problem 14.
- And it asks us what is the solution to the inequality x
- minus 5 is greater than 14?
- Well, to do this, this is just like solving
- any equality or equation.
- What we do to one side, we have to do the other side.
- And we want to get rid of this minus 5, and the best way to
- get rid of a minus 5 is to add 5, so lets add 5 to both sides
- of this equation.
- So 5 plus and then a plus 5.
- And then a 5 plus a minus 5, that's 0.
- That's why we added the 5 in the first place.
- So we're just left with an x on that side.
- We get x is greater than 14 plus 5, which is 19.
- That's choice B.
- Problem 15.
- The lengths of the sides of a triangle are--
- so we have a triangle.
- They tell us the lengths of the sides are y, y plus 1, and
- 7 centimeters.
- They also tell us the perimeter is 56 centimeters.
- The perimeter is equal to 56 centimeters.
- What is the value of y?
- So the perimeter of any shape is just the sum of the sides.
- So y plus y plus 1 plus 7.
- That's the distance around the triangle.
- And that is equal to the perimeter,
- which is equal to 56.
- Let's see, we get y plus y is 2y, plus 1 plus 7 is 8, is
- equal to 56, so you get 2y is equal to what is that?
- 56.
- So if we subtract 8 from both sides of this equation, on the
- left-hand side, we just get 2y.
- On the right-hand side, 56 minus 8, that's 48.
- Divide both sides by 2 and you get y is equal to 24.
- And that is choice A.
- Problem 16.
- Now what do they want us to do?
- All right, I think this is one I should copy and paste.
- OK, now they're telling us-- let me pick a good color.
- Which number serves as a counterexample to the
- statement below?
- So, a counterexample, an example that shows that this
- isn't always true.
- So the statement is all positive integers are
- divisible by 2 or 3.
- So we just have to find a positive integer that is not
- divisible by 2 or 3, by neither 2 nor 3.
- Well, 100 is divisible by 2, right?
- So that just verifies or it's just another example of a
- positive integer that's divisible by one of these two,
- so it's not choice A.
- It's not a counterexample.
- 57, It's not divisible by 2, but it is divisible by 3.
- 19 times 3 is 57.
- So it's not choice B.
- 57 is another positive integer that's divisible by 2 or 3.
- It's divisible by 3, so it's not that.
- 30's divisible by both, so that's definitely not a
- counterexample.
- But here we have 25.
- It's a positive integer and it is neither divisible by 2 nor
- 3, so it disproves the statement.
- So it is a counterexample.
- And so the answer is D.
- Problem 17.
- Let me copy and paste this one.
- Copied it.
- Pasted it.
- All right.
- What is the conclusion of the statement in the box below?
- OK, they say if x squared is equal to 4.
- So if we know that x squared is equal to 4, then we know,
- and we know this from algebra, we could have solved it, that
- x is equal to minus 2 or 2.
- All right, so what is the conclusion of the statement in
- the box below?
- Oh, OK.
- I think I'm reading this too much.
- This is the condition and this is the conclusion.
- They're saying if this has happened,
- then we conclude that.
- So they're actually just asking us to label it.
- This is the conclusion.
- Then x is equal to minus 2 or 2.
- I don't like that question so much.
- My reaction was, OK, if this is the whole statement, what
- can I conclude from it?
- And there's not a lot that I could conclude from it unless
- they told me that this indeed was true.
- But anyway, I don't want to get too complicated.
- They're just saying that if this is true, then we can
- conclude this.
- And they're actually just saying what is the conclusion?
- So what part of this statement can we label as the
- conclusion?
- Well, that's the conclusion, so that's choice D.
- I didn't like that.
- That was more meaning of words than math.
- But anyway, problem 18.
- All right, let me copy and paste this one, too.
- Which of the following is a valid
- conclusion to the statement?
- OK.
- You see, now if a student is a high school band member, then
- the student is a good musician.
- So if you're in the high school band, high school band
- member, then you're going to be a good musician.
- That's what this says.
- And if you compare this to the last question,
- that could be confusing.
- Because a valid conclusion-- so once again, here they're
- giving us a statement and they want us to come to some
- conclusion.
- They might have said what is the
- conclusion of this statement?
- And then you would have said, oh, the
- student is a good musician.
- But that's not what they're asking.
- They're saying, OK, if this whole thing is a statement, if
- we know that this whole thing is true, if they're in the
- high school band, then they're a good musician, which I kind
- of wrote in shorthand here.
- They're asking, OK, if we know that to be true, which of
- these statements can we conclude from that?
- Statement A says, all good musicians are
- high school band members.
- Well, that doesn't tell us that.
- It doesn't say that all good musicians-- the good musicians
- could be this circle and the high school band members would
- be a subset of it.
- High school band, and this could be good musicians.
- So right, all high school band members are good musicians,
- but there could be people out here who are good musicians
- who aren't high school band members.
- So it's not choice A.
- A student is a high school band member.
- Well, no.
- High school band members, they didn't tell us that all of the
- students are high school band members or the high school
- band is comprised of all of the students.
- The student body might be like this circle.
- Some of them might be in the high school band.
- Some of them aren't and are good musicians.
- Some of them aren't and aren't good musicians.
- So, I mean, you don't know whether someone is a high
- school band member just by being a student.
- All students are good musicians.
- Once again, you know, this could be an example.
- There could be people out here who are students, the yellow
- could be students.
- There could be people who are students
- who aren't good musicians.
- This statement doesn't in any way constrain that.
- So let's get rid of that.
- All high school band members are good musicians.
- Well, let's look at this Venn diagram.
- I mean, that's almost like a restating of what they
- already told us.
- All high school band members have to be good musicians
- because they told us, if you're in the high school
- band, then you're a good musician.
- So it's almost like repeating the same statement twice, but
- the answer is D.
- Problem 19.
- Let me erase this one.
- This looks like another one that I'm going to
- have to copy and paste.
- OK, copied it.
- OK, let me do a dark color.
- The chart below shows an expression evaluated for four
- different values of x.
- When x is equal to 1, x squared plus x plus 5, right.
- 1 plus 1 plus 5 is 7.
- When x is equal to 2, then 2 squared plus 2 plus 5 is 11.
- Fair enough.
- Josiah or Hosiah, I don't know how to pronounce that,
- concluded that for all positive values of x, x
- squared plus x plus 5 produces a prime number.
- Which value of x serves as a counterexample to prove
- Josiah's conclusion false?
- So a counterexamples says-- he says that whenever I put any
- positive number here, I get a prime number here.
- We have to say which one proves him false?
- If you put 5 there, what is-- 5 squared is 25 plus 5 plus 5,
- and what is that equal to?
- That's equal to 35.
- So that's a counterexample right there.
- If x is equal to 5, I produce 35, which is not a prime
- number, not prime.
- So clearly, his statement, his conclusion, was incorrect,
- that when you put a positive number here, it doesn't always
- produce a prime number.
- That was his conclusion.
- This one isn't prime.
- So statement A is a counterexample, or the number
- 5 is a counterexample here.
- Next problem, problem 20.
- OK, this is one of those where we have to pinpoint the
- incorrect step.
- John's solution to an equation is given below.
- For which step of real numbers did-- oh, which property of
- real numbers did John use for step 2?
- OK, so we don't even have to look at step 1.
- We just have to say, OK, how did he get
- from there to there?
- So when you look at this, when you say x plus 2 times x plus
- 3 is equal to 0, you're saying some number x plus 2 times
- some other number x plus 3 is equal to 0.
- So that's like saying some number times some other number
- is equal to 0.
- So that means that one of those numbers or both of them
- have to be equal to zero, right?
- Because the only way to get zero is if one of these or
- both of these have to be equal to zero.
- And that's where he gets this conclusion that either x plus
- 2 has to be 0 or x plus 3 is 0.
- They're both zero.
- So let's see how they-- what do they call it?
- Multiplication property of equality.
- I don't even know what that means.
- Zero property of multiplication?
- This seems like the closest one for me so far.
- That anything times zero is zero Or for two things to be
- multiplied to equal zero, at least one of
- them has to be zero.
- And that's what this is right here.
- Commutative property of multiplication?
- No, that's not it.
- Distributive property of multiplication over addition?
- No, we're not doing anything like that.
- If we went from this step to that step where we're
- multiplying it, that might have been, because you're
- really just doing the distributive property, but I
- don't want to confuse you.
- Yeah, we're just saying that if two numbers when you
- multiply them equal zero, one of them have
- to be equal to zero.
- And I guess the label for that is the zero product property
- of multiplication.
- Anyway, see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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