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: Graphing Inequalities 21-26, graphing inequalities and testing assertions
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- We're on problem 21.
- Stan's solution to an equation is shown below.
- All right, and I haven't looked at it yet, but let's
- just see what they ask us.
- Which statement about Stan's solution is true?
- So let's just work through it and see if he got it right or
- if he made a mistake.
- So let's see.
- Let's think about it how we would do it.
- So if we had n plus 8 times n plus 20 is equal to 110.
- So the first thing you want to do is you'd want to distribute
- this 8 times n plus 20.
- You don't want to multiply this whole n plus 8.
- Remember, you're doing order of operations, so it's
- multiplication first. Let's do that.
- So you get the n plus and you distribute this 8.
- So it's 8n plus 160 is equal to 110.
- 8 times n plus 8 times 20 is 160.
- And already I see a discrepancy
- between ours and his.
- We have n there.
- We have plus 8n there.
- But where he has a 20, we have 160.
- He didn't distribute the 8.
- 8 times n plus 20 is 8n plus 160.
- So that's wrong.
- So he made a mistake in step 1.
- Unfortunately, he got all that other work that he had to do
- just because of his-- it's all going to be wrong because of
- his mistake.
- Problem 22.
- When is the statement true?
- The opposite of a number is less then the original number.
- Well, I mean, if I have a negative number, the opposite
- of a negative is positive.
- So it's not going to be less.
- But if l positive number, then the opposite of it is
- negative, which is less than it.
- Because a negative is less than a positive.
- Or if we say that x is greater than 0, then minus x is going
- to be less than x.
- If we say x is positive, then definitely.
- You know, minus 3 is less than positive 3.
- So the statement is true for positive numbers.
- So that's C.
- It's definitely not never true.
- I mean we showed a case where it's true.
- If I have positive 3, negative 3 is less than that.
- Statement B.
- The statement is always true.
- Well no, if we start with negative 3, the opposite is
- going to be larger.
- The opposite's going to be positive 3.
- And D.
- The statement is true for negative numbers.
- No, it's not.
- Negative 3, the opposite is 3.
- And that's larger than negative 3.
- Problem 23.
- What is the y-intercept of the graph?
- OK, let me see.
- So 23.
- They say 4x plus 2y is equal to 12.
- So if you were to graph this line, not that I'm going to--
- the y-intercept is when it intersects the y-axis.
- So it's that point there.
- It's what's this y-coordinate.
- Well the x-coordinate, it's when x is equal to 0.
- So it's going to be 0 comma some y-intercept.
- So the easiest thing to do is say when x is equal
- to 0, what is y?
- So when x is equal to 0, this 4 times 0 is equal to 0.
- So that just becomes 0.
- So you get 2y is equal to 12.
- y is equal to 6, and that's choice C.
- All they want to know is, of this line, when x is equal to
- 0, what's y?
- That's what the y-intercept is.
- Next problem.
- OK, they've drawn us a graph.
- They say-- let me copy and paste this one.
- OK.
- Copied it.
- OK, which inequality is shown on the graph below?
- So first of all, let's figure out the equation of
- this line right now.
- So what's its y-intercept, first of all?
- So when x is equal to 0, y is equal to minus 1.
- If we put it in slope y intercept form, the equation
- of really, any line is y is equal to mx plus b.
- Where m is the slope and b is the y-intercept.
- Here, when x is 0, y is negative 1.
- So the y-intercept is minus 1.
- So b is minus 1.
- So we know that y is equal to mx minus 1.
- And now we just have to figure out the slope.
- And the slope is just change in y for a given change in x.
- So if we say change in y for a given change in x is equal
- to-- let's see.
- When I increase x by 2, I increase y by 1.
- So change in y is 1 when the change in x is 2.
- And you could say when you increase x by 4-- I'm going
- four spots-- I increase y by 2.
- So you could have done that.
- You could have said change in y over change in x is 2/4,
- which is also equal to 1/2.
- But either way, we now know the slope.
- The slope is change in y over change in x.
- So the equation of the line is y is equal to 1/2 x minus 1.
- That's the equation of this line.
- Now they want to know what the equation of the inequality is.
- So first of all, they're including-- they have this
- grey area and they put this line in bold.
- So that means that we're including the line.
- If they had drawn a dotted line here that means we're not
- including the line.
- But since they're including the line, it's going to be
- either greater than or less than or equal to.
- That equal to is because we're including the line.
- But just think about it.
- So our choices are y is greater than or equal to 1/2 x
- minus 1 or y is less than or equal to 1/2 x minus 1.
- Now think about for any given x.
- Let's say when x is equal to 2.
- When x is equal to 2, if you put it into this equation you
- get y is equal to 0 and that's this point on the line.
- Now is the grey area all y values greater than 0 or all y
- values less than 0?
- Well clearly, it's all y values greater than 0.
- It's all the y values above the y value
- dictated by this equation.
- The equation of this inequality or this area is y
- is greater than 1/2 x minus 1.
- And just a very easy way to eyeball it is OK, they're
- including the line, so I'm going to have this equal
- then-- it's going to either be greater than or equal to or
- less than or equal to, and since it's the area above the
- line it's going to be greater than or equal to.
- And that is choice D.
- Next problem, problem 25.
- This is another one where I think it makes sense to copy
- and paste it.
- It's a big one.
- OK.
- Which best represents the graph of y id
- equal to 2x minus 2.
- So the slope is 2.
- Its positive 2 and its y-intercept is minus 2.
- So the y-intercept is easy to eyeball.
- Here the y-intercept is minus 2.
- That's right.
- Here it's minus 2 as well.
- In both of these, the y-intercept is plus 2.
- So neither of these are going to, right?
- It intersects the y-axis at positive 2.
- But we know that the y-intercept is minus 2.
- So we know-- whoops.
- Where'd I go?
- There we go.
- So we know our choices are one of the top two.
- It's either A or C.
- And which of these has a positive slope,
- positive slope of 2?
- Well in choice A when I go to the right by 2, I go up by 4.
- So change in y over change in x.
- I go up by 4, when I go up in x by 2.
- I go up in y by 4 when I go up in x by 2, so the slope is 2.
- So this has a positive slope of 2, so that's choice A.
- But just for seeing why choice C doesn't work or figuring out
- the slope, think about it this way.
- Let me start at a random point here.
- If I increase x value by 2, what's
- happening to my y value?
- Is it increasing by 2 or is it decreasing by 2?
- Well it's decreasing.
- Well it's decreasing by 4.
- It's decreasing at twice the rate.
- It's decreasing by 4.
- So in this case, change in y over change in x.
- Whenever x is positive 2, we increased by 2 here.
- What happened to y.
- It went down 4.
- So here the slope is minus 2.
- And you can look at that.
- You can kind of eyeball it because it's going from the
- top left to the bottom right, so it's going to
- be a negative number.
- But the easiest way I always do is to draw these arrows and
- say, OK, when I increase x, y is decreasing.
- So that's going to be a negative slope.
- Let's do it.
- So the answer was A.
- Next problem.
- OK, so they're giving us another one of these where
- they shaded a graph.
- These are good.
- Thank God I have the cut and paste feature.
- So they ask us, which inequality does the shaded
- region of the graph represent?
- So once again, it'll be good just to figure out what the
- equation of this line is.
- We could immediately say, OK, the y-intercept is 2.
- And what's its slope?
- When x increases by 1, what happens to y?
- y is going down by 3.
- So the y-intercept is equal to 2.
- We figured that out. b is equal to 2.
- The slope is equal to change in y over change in x.
- When x increases by 1, the slope-- sorry.
- The change in y is minus 3.
- It goes down by 3.
- So the slope is equal to minus 3.
- So the equation of this line is y is equal to
- minus 3x plus 2.
- mx plus b.
- And if I look at the choices, already it looks a little bit
- different than that, but we'll get to that.
- But let's see what the inequality is.
- So first of all, it's going to be either greater than or
- equal or less than or equal.
- And we say the or equal because they
- filled in the line.
- So let's think about it.
- If we pick any x, say when x is equal to 2, then according
- to this equation, according to this line, y would be
- equal to minus 4.
- Now, are we including the y's that are greater than that
- value as well or are we including the y's that are
- less than that value?
- Well, we're including the ones that are less than that point
- for a given x.
- So the shaded region is y is less than or equal to
- minus 3x plus 2.
- And you can eyeball this.
- You can kind of say, oh, this is below the graph, so it's
- less than or equal.
- And that equal was because they shaded in the line.
- If it was a dotted line it would just be less than.
- Once again, this isn't the exact format
- that they have here.
- They put all the x's and the y's on the same side, and we
- can do that.
- Let's just add 3x to both sides of this equation and you
- get 3x plus y is less than or equal to.
- When you add 3x on the right-hand side that
- just becomes 0.
- So less than or equal to 2.
- And that is 3x plus y is less than or equal to positive 2.
- And that is choice A.
- And I'm all out of time, see you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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