Absolute value
CA Algebra I: Number Properties and Absolute Value 1-7, number properties and absolute value equations
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- We're now going to take the California Standards Test,
- Algebra I released questions.
- In the last series, I had done the Algebra II.
- I guess I'm going in reverse order.
- Let me copy and paste this first question because I think
- it's good to see the whole thing.
- So let me see, I have copied it.
- Let me move this pointer all the way up, and
- then, there we go.
- All right.
- And they're asking us is the equation 3 times 2x minus 4
- equal to minus 18 equivalent to 6x minus 12 is equal to 18?
- So let's think about this.
- If we just distribute this 3, what do we get?
- 3 times 2x is 6x.
- 3 times minus 4 is minus 12.
- And that, of course, is equal to minus 18.
- So sure, they're the same thing.
- If you just distribute the 3 over the 2x minus 4, you get
- 6x minus 12.
- So the answer is definitely yes.
- It's not the no one down here.
- And it says yes, the equations are equivalent by the
- associative?
- No.
- Communicative?
- No.
- The equations are equivalent by the distributive property?
- [FIRE TRUCK SIREN]
- There's some type of a fire truck going on outside.
- Let's see.
- Where was I?
- Oh, yes.
- Yes, the equations are equivalent by the distributive
- property of multiplication over addition.
- Right, that's that.
- We distributed this 3 over the 2x minus 4.
- And they say over addition because you could view this as
- a plus minus 4.
- Addition and subtraction is really the same thing when you
- think of the distributive property.
- Anyway, let's do the next problem.
- The next problem I can just write out.
- This is problem number 2.
- They say the square root of 16 plus the cube root
- of 8 is equal to?
- Well, what's the square root of 16?
- And when you just have a square root there, you might
- say, maybe it's plus or minus 4, but when they write it this
- way is means the principal root, so it's just plus 4.
- They would write a plus or minus out front if they wanted
- you to get the negative square root.
- So it's 4 plus-- now, what to the third power is equal to 8?
- Well, 2 to the third power is equal to 8, right?
- So we could write 2 to the third is equal to 8.
- That's the same thing as saying the cube root of 8 is
- equal to 2.
- You could also view this as 8 to the 1/3 power.
- Anyway, the cube root of 8 is then 2, so 4 plus 2 is equal
- to 6, and that is choice B.
- Problem 3.
- Let me scroll down a little bit.
- OK, and they want to know-- I could copy and
- paste the whole thing.
- There we go.
- And they want to know which expression is equivalent to x
- to the sixth times x squared?
- So x to the sixth times x squared, they
- have the same base.
- When you're multiplying both of these expressions, we can
- add the exponents.
- So that is equal to x to the-- 6 plus 2 is 8.
- That's not one of the choices here, so we have to say, which
- of these also are the same thing as x to the eighth?
- And so which two exponents when I add them is equal to 8?
- 4 plus 3 is equal to 7.
- 5 plus 3, this is equal to x to the eighth as well.
- So that is choice B.
- Next problem, problem 4.
- All right, let me-- this is another one where I'll copy
- and paste it.
- All right.
- They want to know which number does not have a reciprocal?
- So the reciprocal of negative 1, that's just 1 over negative
- 1, which is equal to negative 1.
- The reciprocal of 0, that's what?
- 1/0, which is not defined.
- So the choice is B.
- 0.
- We don't know what 1/0 is.
- Maybe that's a project for you to think of
- what it should mean.
- And, of course, these have reciprocals.
- 1 over 1/1,000 is just equal to 1 times 1,000 over 1, which
- is equal to 1,000.
- And the reciprocal of 3 is, of course, 1/3.
- Next problem.
- They say-- so there's a lot of terminology here, but I guess
- that's good.
- So they want to know-- let me just copy.
- Maybe I'll do the next one, too.
- OK.
- I could probably just do it up here.
- All right.
- They want to know, what is the multiplicative inverse of 1/2?
- So essentially, what can I multiply 1/2 by
- and then get 1?
- It's the same thing as saying what's the inverse of 1/2.
- So if I multiply by 1/2 by-- well, the inverse of 1/2, I'd
- say 1 over 1/2.
- That's the same thing as 1 times 2/1,
- which is equal to 2.
- Or another way to think of it is 2 times 1/2 is equal to 1.
- So the multiplicative inverse of 1/2 is just 2.
- That's choice D.
- Problem 6.
- What is the solution for this equation?
- All right, sometimes these absolute value signs can
- appear daunting, but you just have to
- think it through logically.
- If the absolute value of 2x minus 3 is equal to 5, that
- tells us what?
- That means that 2x minus 3 is equal to 5, right?
- Because inside the absolute value is equal to 5, then the
- absolute value of 5 is equal to 5.
- So that's fair enough.
- But what could 2x minus 3 also be equal to?
- What happens if 2x minus 3 within the absolute value sign
- is equal to minus 5?
- Well, then you would take the absolute value of that and you
- would get 5, right?
- So 2x minus 3 could also be equal to minus 5.
- When you see this absolute value sign, you say, OK,
- whatever's inside the absolute value is either 5 or negative
- 5 because we're taking the absolute value of it to get 5.
- So we just solve both of these equations.
- If you add 3 to both sides of this one, you get
- 2x is equal to 8.
- x is equal to 4.
- On the second one, you add 3 to both sides.
- You get 2x is equal to-- minus 5 plus 3 is minus 2.
- x is equal to minus 2 divided by 2 is negative 1.
- So x could be equal to 4 or x could be equal to negative 1.
- And that is choice C, x is negative 1 or x is equal to 4.
- Next problem.
- The Algebra I ones go faster than the Algebra II problems.
- Those tend to be hairier.
- Let me clear all of this.
- I'll just write this one down.
- They say what is the solution set for the inequality 5 minus
- the absolute value of x plus 4 is less than or
- equal to minus 3?
- So at first, this is really daunting.
- I can't even do that logic that I did last time because I
- have that 5 out there.
- But let's think of it this way.
- Let's try to simplify it, so we have just the absolute
- value of something is less than or equal
- to something else.
- So one thing we can do is, if we want to get rid of this 5,
- remember, what we do to both sides of an equation or
- inequality-- whatever we do to one side of an equation or an
- inequality, we do to both sides.
- So let's subtract 5 from both sides of this equation.
- If you subtract 5 from the left side, this 5 disappears.
- I'm just going to do minus-- let me write that out.
- Minus 5 plus, and I'm going to do a minus 5 there.
- That's a plus.
- So minus 5 plus 5 is 0, so I'm just left with minus absolute
- value of x plus 4 is less than or equal to-- now, what's
- minus 3 minus 5?
- That's minus 8.
- All right, now this next step, this is something-- maybe it
- wasn't obvious for you and putting the inequality there--
- you know, if this was an inequality, you would just
- say, OK, I'm going to multiply or divide both sides by
- negative 1 to get rid of the negative signs.
- But one thing you have to remember, whenever you
- multiply or divide both sides of an inequality by a negative
- number, you have to switch the inequality.
- So if this is true, then if I'm multiplying both sides of
- this by negative 1, so negative 1 times negative x
- plus 4, I'm going to switch the inequality, so that's
- going to become greater than or equal to negative 8.
- And I did the negative 1 on this side, so I have to
- multiply it times negative 1 on that side.
- And so this negative cancels out that negative, so we're
- just left with x plus 4 is greater than or equal to--
- negative 8 times a negative 1 is equal to 8.
- Now we can just use the logic that we had
- from that last problem.
- This tells us what?
- This tells us that the magnitude of x plus 4 is
- greater than or equal to 8.
- Let me draw a number line here because I really want you to
- get the intuition of what magnitude means.
- So if that's the number line and you can view magnitude as
- kind of the distance from, or the absolute value, you can
- kind of view it as the distance from 0, right?
- So if this is 0 right here and this is positive 8 and this is
- minus 8, the absolute value of whatever this quantity was is
- greater than 8.
- That means its distance from 0 has to be greater than 8.
- You can just say distance from 0 of this number has to be
- greater than 8, greater than or equal to 8.
- That means that this number is definitely going to be greater
- than or equal to positive 8.
- On the number line, it would be all of
- those numbers, right?
- Or, remember, we're saying the magnitude, so we don't care
- about the direction.
- The magnitude has to be greater than positive 8, so it
- also includes the negative numbers less than negative 8.
- And why does that make sense?
- Well, take negative 9.
- What's the absolute value of negative 9?
- The absolute value of negative 9 is greater than 8 because 9
- is greater than 8, so any number to the left of negative
- 8 or to the right of positive 8.
- So what does that tell us about this equation?
- So that means that-- well, the easy one is x plus 4 could be
- greater than or equal to 8.
- So let's write that down.
- Let me write it here.
- x plus 4 greater than or equal to 8.
- And that takes into consideration that the
- magnitude is greater than or equal to 8 there.
- Or x plus 4 less than or equal to minus 8.
- That's the magnitude to the left of this
- negative 8 right there.
- And now we solve it.
- And it's very important to think of absolute value in
- these terms. Otherwise, it can become very confusing and you
- start testing numbers.
- But if you really just visualize the number line and
- you think of absolute value as distance from 0, magnitude of
- the distance from 0, you say, oh, the distance from 0 has to
- be greater than or equal to 8, so that means my number has to
- be-- this thing has to be less than or equal to minus 8 or it
- has to be greater than or equal to positive 8.
- So let's solve.
- x plus 4 is greater than or equal to 8.
- Subtract 4 from both sides, so you get x is greater than or
- equal to 4.
- I just subtracted 4 from both sides.
- Subtract 4 from both sides here, you get x is less than
- or equal to minus 12.
- So the solution here is x is greater than or equal to 4 or
- x is less than or equal to minus 12, and
- that is choice D.
- Anyway, I'll 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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