Algebra II
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California Standards Test: Algebra II
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California Standards Test: Algebra II (Graphing Inequalities
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CA Standards: Algebra II (Algebraic Division/Multiplication)
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CA Standards: Algebra II
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Algebra II: Simplifying Polynomials
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Algebra II: Imaginary and Complex Numbers
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Algebra II: Complex numbers and conjugates
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Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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Algebra ||: Conic Sections
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Algebra II: Circles and Logarithms
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Algebra II: Logarithms Exponential Growth
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Algebra II: Logarithms and more
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Algebra II: Functions, Combinatorics
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Algebra II: binomial Expansion and Combinatorics
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Algebra II: Binomial Expansions, Geometric Series Sum
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Algebra II: Functions and Probability
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Algebra II: Probability and Statistics
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Algebra II: Mean and Standard Deviation
California Standards Test: Algebra II (Graphing Inequalities 6-8 (understanding and graphing inequalities)
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- We're on problem number six and I actually figured out how
- to cut and paste from the PDF file to my little writing
- board, so we actually have access to the problem.
- So it says: What system of inequalities best represents
- the graph shown below?
- So we want to figure out what inequalities represents this
- shaded area.
- So a good place to start is just to figure out what these
- dotted line graphs, what the equations of them look like.
- So this one, let's do the hard one first, right?
- This is at an angle.
- So whenever I look at a line, I like to think what's it's
- slope and what it's y-intercept, right?
- And then you could just write it in the form y is equal to
- mx plus b, where b is the y-intercept
- and m is the slope.
- So let's try to figure out the equation of this line up here,
- this dotted line.
- So what's its slope?
- So when I move over 1 and x, when I move over to the right,
- when x increases by 1, or let's say when x increases by
- 2, how much does y increase?
- Well, y increases by 2 as well.
- Or whatever we increase x by.
- If x increases by 1, then y increases by 1.
- So the slope is 1, right?
- Slope is just change in y over change in x.
- So the slope of this first line is 1.
- So m is equal to 1, for at least this first line up here.
- And what's it's y-intercept?
- The y-intercept is just what does the graph equal when x is
- equal to 0 or where does it intersect the y-axis?
- Well, it intersects the y-axis right here at y is equal to 1,
- so b is equal to 1.
- So the equation of this dotted line up here is y is equal to
- mx, the slope times x, we figured out the slope was 1,
- so that's x plus the y-intercept, which we also
- figured out was 1.
- So that's the graph of that line, right?
- But notice, the shaded region is below that line.
- And it also does not include the line, right?
- That's why they drew a dotted line.
- If it was a solid line, that means that we're including the
- line itself.
- But the shaded region does not include the line.
- But it's below it.
- Think of it this way: For any given x-- let's pick this x.
- Let me do it in a different color.
- Let me do it in blue.
- If I were to pick x is equal to 1, this is the point y is
- equal to x plus 1.
- That's the equation.
- Now, my question to you is y above y is equal to x plus 1?
- Is the shaded region y is greater than x plus 1, which
- would be that region?
- Or would it be y is less than x plus 1,
- which is this region?
- Well, clearly, it is y is less than x plus 1.
- You pick any x, x is equal to 3.
- This is what y would be equal to. y would be equal to 3 plus
- 1 if you picked this choice.
- But the y's that satisfy the grey area are always less than
- this point.
- So at least the grey area relative to this equation, we
- can write the inequality y is less than x plus 1.
- And we know it's not less than or equal to.
- It would be less than or equal to if they had filled in this
- line, if this was part of the area, but it's not.
- They drew a dotted line.
- So that's why we know it's less than, right?
- If the grey area was up here it would be greater than.
- You just pick an x, you say, oh, the y's less than the
- line, not the y is greater.
- But we're not done, right?
- Because this would be the solution set if all of this
- area was grey.
- But it's only grey up to a certain point.
- It's only grey up to this line down here.
- This line is easier.
- This line has no slope, right?
- No matter what my change in x, my change in y is 0.
- So what's the equation of this?
- The equation of this line is just slope is 0.
- So y is equal to 0x plus the y-intercept.
- Well, the y-intercept is minus 2.
- So the equation of this line is y is equal to minus 2.
- Let me do it in the purple.
- I'll just write it here.
- That equation is y is equal to minus 2, Which makes sense.
- For all x, y is equal to minus to 2.
- Now, my question to you is this grey area.
- For any given x, the y is greater than the line or the y
- is less than the line?
- Well, let's pick this x.
- When x is equal to 3, y is equal to negative 2.
- That's this line.
- Now, is the grey area-- the y's for that x-- above the
- line or below it?
- Well, it's above the line, right?
- So if we look at it from the point of view of this
- boundary, we say that y has to be greater than minus 2.
- That's the shaded area.
- y is equal to negative 2 is this dotted line's equation.
- The shaded area, at least relative to this line, is y is
- greater than negative 2.
- If I just had this by itself, I would shade up everything
- above y is equal to negative 2, right?
- But I want just the combination of both of these.
- So essentially, we have to say that y-- let me do it in a
- nice, vibrant color.
- Since the shaded area is below this line, we know that y is
- less than x plus 1.
- And since the shaded area is above this line, we know that
- y is greater than negative 2.
- Where could I write that?
- I'll write it right here.
- So which of the choices satisfy that?
- Let's see.
- Whoops, I went to the next problem.
- So we know that y is greater than negative 2.
- So these can't-- let me just cross those out.
- That and that can't be the answer.
- And then we also know that y is less than x plus 1.
- So we know that this is the answer.
- And frankly, if you just looked at the choices, you
- really didn't have to figure out the slope and y-intercept.
- They're essentially giving you the equations of the two
- lines, right?
- You say, like, oh, one is x plus 1 and one is negative 2,
- and I just have to figure out whether y is greater than or
- less than each of those lines.
- And you can say, OK, relative to the y equals negative 2,
- we're definitely greater than negative 2, so I'm one of
- these two choices, and relative to y is equal to x
- plus 1, we're definitely less than that line, so we have to
- be choice b.
- Next problem.
- Which point lies in the solution set for the system?
- So you essentially just have to say which coordinates
- satisfy both of these equations?
- And frankly, you could graph them and you could draw the
- area like we did in the last one, or you could just try out
- the numbers, right?
- Let's just try the numbers.
- That never hurts.
- So minus 4, 1.
- Let's see if it satisfies it.
- So remember, the first coordinate is x, the
- second one is y.
- So 2y would be 2 times minus 1 minus x.
- So minus minus 4.
- Now, is that greater than or equal to minus 6?
- Well, this is minus 2 plus 4.
- And this is what?
- Minus 2 plus 4 is equal to 2, which is definitely greater
- than or equal to minus 6.
- So it satisfies the first equation.
- Let's see if it satisfies the second equation.
- I'll do it in another color and I'll do it up in the black
- area up here.
- So 2 times y.
- So it's 2 times minus 1-- y is minus 1-- minus
- 3 times minus 4.
- And they want to know is that less than minus 6?
- So you get minus 2.
- 3 times 4 is 12, so it's plus 12.
- Negative times a negative is a positive, so that equals plus
- 10, which is not less than negative 6.
- So the first one does not work.
- That is not the answer.
- Let's try the second one.
- 3 comma 1.
- So y is 1.
- So it becomes 2 times 1, which is just 2, minus 3.
- Is that greater than or equal to negative 6?
- So you get negative 1 is definitely greater than
- negative 6, so this satisfies the first equation.
- Does it satisfy the second one?
- 2 times y.
- 2 times 1 is 2, minus 3 times 3.
- Is that less than minus 6?
- So you get 2 minus 9.
- Is that less than minus 6?
- You get minus 7 is less than minus 6.
- So that is true.
- So choice b works.
- You don't have to go any further.
- That point lies in the solution set, which just means
- it satisfies both of these inequalites.
- Next problem.
- Scroll down a little bit.
- Which system of linear equalities is represented by
- this graph?
- So now we can use some of the skills we used the last time.
- I don't have much space to draw all of them.
- So let's just to figure out the-- let's see, do they gives
- us-- no, they don't even give us the equations because all
- the equations are different, so we have to figure out the
- equations of these two graphs.
- So let's do this top one first. So let's
- figure out its slope.
- When you increase-- and you can just eyeball this one.
- When you increase x by 3, y increases by the
- same amount, right?
- So change in y over change in x would be 3 over 3.
- Whatever you change x by, y increases the same amount.
- So the slope of this first one at least-- let me do it in a
- color you can see over black-- the slope is equal to 1.
- And what's its y-intercept?
- Well, you just look at where did it intercept the y-axis?
- Right there.
- So the y-intercept is equal to minus 2.
- So this first equation right here is y is
- equal to x minus 2.
- Fair enough.
- And this grey area, is it above or below it?
- It's below it.
- You pick any x.
- For any x, this is y is equal to x minus 2.
- The grey area is for all the y's that are
- less than it, right?
- And notice, now the line is shaded in.
- So the line is part of the solution set.
- So at least relative to this, we know that the grey area is
- all y's that satisfy y is less than or equal to x minus 2.
- Less than because it's below it, and it's equal to because
- they actually drew in the line.
- But now we have to figure out the lower bound.
- What's the equation of this line?
- So let's think about it a little bit.
- This one's a little more interesting.
- If I increase x by 1, so if I'm going from this point,
- what happens to y?
- y goes down by 2, right?
- 1 minus 2.
- If I increase x by 2, y goes down by 4, right?
- Minus 4.
- So at least in this case, the slope, change in y, is minus 4
- whenever change in x is positive 2.
- Or it could've been change in y is minus 2 whenever change
- in x is plus 1, right?
- So it equals minus 2.
- And that makes sense.
- For every 1 you go over, the slope goes down 2.
- And what's it's y-intercept?
- Let's see, y-intercept is just right there. y is equal to 3.
- So b is equal to 3.
- So this second graph is y is equal to minus 2x plus 3.
- And then, once again, the shaded area includes this, but
- we have to be greater than or equal to this graph.
- We have to be greater than, because if you pick any x, the
- y's that satisfy the shaded area are equal to the graph or
- greater than for any x.
- So there we have our inequality.
- So this one, this area, you could kind of say is y has to
- be greater than or equal to minus 2x plus 3.
- That's from this boundary.
- And we also know that y has to be less than or
- equal to x minus 2.
- Now, hopefully, that's one of the choices.
- If we haven't a made a careless mistake.
- Let me see.
- All right.
- y is greater than or equal to-- OK, so
- this is not the answer.
- Neither of these are the answers.
- That's not the answer.
- Let me cross that out.
- y is greater than or equal to-- we have
- minus 2x plus 3, right?
- Let me make sure I got that right.
- Minus 2x plus 3.
- Yeah, let me see.
- Right, so they're tricky.
- So the slope on that second line was minus 2x plus 3.
- They got the second part right.
- Maybe I made a careless mistake, but I don't think
- this is right either.
- Let's see if we can rearrange.
- So these other two, they put the x's and the y's on the
- same side, so let's do that.
- That top equation, we can rewrite it.
- Add 2x to both sides.
- You get 2x plus y is greater than or equal to 3.
- And the second equation, if we were to subtract x from both
- sides, you get y minus x is less than or equal to minus 2.
- And now this is interesting.
- So there's no minus 2 on the right-hand side here,
- but what can we do?
- If we multiply both sides of this equation by
- minus 2, what happens?
- Sorry, multiply both sides of the equation by minus 1.
- You get minus y plus x.
- And when you multiply an inequality by a negative
- number, multiply or divide, you switch the inequalites.
- So it's greater than or equal to-- and we're multiplying by
- negative 1-- 2.
- So it becomes x minus y is greater than or equal to 2.
- So our two equations turn into 2x plus y is greater than or
- equal to 3, which is this one.
- And we get x minus y is greater than or equal to 2,
- which is that one.
- And we're done.
- It's choice d.
- 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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