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: Simplifying Expressions 8-13, simplifying expressions and word problems
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- All right, we're on problem number 8.
- They ask us
- which equation is equivalent 5x-2(7x+1)＝14x
- So I'm guessing they just want to simplify this a little bit
- and see if we get to one of these choices.
- So let's do that.
- So we start with 5x-2(7x+1)＝14x
- This is I think the most obvious thing to simplify.
- This 2(7x+1) or we can even say -2(7x+1)
- so then this becomes 5x plus
- I'm just going to distribute the -2 times all of this.
- Plus -2 times 17x is minus 14x.
- minus 14x
- And then -2 times 1 is minus 2 is equal to 14x.
- And let's see, on all of these choices
- they have 14x on the right hand side.
- So they just want us to simplify this.
- So this simplifies to 5x, plus -14, so that's -14x
- minus 2 is equal to 14x.
- So we have 5x-14x.
- What's 5-14?
- It's what? -9, right?
- -9x-2＝14x
- That is choice A.
- Now, one thing that maybe I realized,
- I kind of skipped a step.
- We could've just...
- Let me just do this
- just so that you understand this step that I did here.
- We could have just said minus
- and then distribute the 2
- So we could have right there said
- 5x minus positive 2 times 7x, 14x, plus 2 times 1, plus 2.
- I'm just doing the left-hand side.
- And then we could have distributed the minus sign.
- So you get 5x-14x-2.
- Either way, it would have gotten us to this point and then
- we could have simplified here and gotten choice A.
- Next problem.
- Which equation is equivalent to
- OK, so another case where
- I'm guessing they just want us
- to simplify these equations
- So let me write that down.
- 4(2-5x)＝6-3(1-3x)
- All right, let's distribute this 4 first.
- So you get 4 times 2 is 8.
- 4 times -5x is -20x.
- -20x is equal to 6 minus--
- maybe you might have been tempted to say,
- oh, 6 minus 3, that's 3, and then distribute it.
- Remember, order of operations.
- Multiplication comes first.
- So we have to multiply the 3(1-3x)
- before we can deal with the 6.
- Well, on the other side
- there was just the multiplication first.
- All right.
- So here, 3(1-3x), so it'll be 3.
- I'm just going to do
- minus a positive 3 times each of these.
- So 3 times a -3 is -9x.
- And now I can distribute this minus sign.
- So let's see, on the left-hand side I have
- let me do it in a brighter color.
- The left-hand side I have 8-20x＝ 6,
- and then I'm going to subtract this whole thing.
- So you can view it as
- distributing a -1 times both of these.
- So 6 plus -3, or just 6-3,
- and then a minus times a minus plus 9x.
- All right.
- Now let's see what we could do to help simplify this.
- What happens if we add 20x to both sides?
- Then on the left-hand side, it goes away
- so we're just left with 8 is equal to
- well,let me just do that.
- 6 minus 3.
- I added 20x, so plus 29x.
- I really should have simplified this 6 minus 3
- but I didn't want to do too much in one step.
- So 8＝3+29x,
- and then if we subtract 3 from both sides here
- we get 8 minus 3 is 5.
- That 3 goes away and you get 29x
- and that is choice C. 29x＝5.
- Next question, problem number 10.
- And I should be careful that you don't think that
- all of this stuff is for this problem.
- The total cost, c, in dollars
- of renting a sailboat for n days
- is given by this equation.
- If the total cost was $360
- for how many days was the sailboat rented?
- So they're giving us c. c is $360.
- So they're telling us that $360, that's the total cost,
- and that's going to be equal to
- 120 plus 60 times the total days rented, times 60n.
- And that's what they want us to figure out, n.
- How many days was the sailboat rented?
- Let's see, so if you should subtract 120
- from both sides of this equation
- you get 240 is equal to 60n.
- I'll go here.
- And if you divide both sides of this by 60
- 240 divided by 60
- that's the same thing as 24 divided by 6
- so that's 4.
- And then 60n divided by 60 is, of course, 1
- so that is n.
- So the sailboat was rented for 4 days.
- All right, problem number 11.
- And I cut and pasted part of it
- but they want us to determine
- what was the first incorrect step
- that they did in this problem?
- So let's see if we can determine that.
- So let's see.
- Solve 3(x+5)＝2x+35
- Fair enough.
- So in this first step they distributed 3*x 3x5
- Right, 3x+15＝2x+35
- so step 1 looks good to me.
- Let me do that in a darker color.
- Step 1 looks good.
- They distributed the 3.
- Now let's see what they did in step 2.
- Let's see, if I wanted to solve this
- I would subtract 2x from both sides,
- and it looks like that's what they attempted to do
- because they got rid of this 2x, right?
- On the right-hand side
- they went from 2x+35 to just 35
- so somehow they got rid of this 2x
- and the only way to get rid of it
- is to subtract 2x from both sides.
- Right? Subtract 2x from both sides
- Now, if I subtract 2x from both sides, sure enough
- the right-hand side will just be left with the 35.
- The left-hand side will have the 15, of course.
- And then I have -2x+3x.
- Well, -2x+3x, that's x.
- So this 5x, that shouldn't be there.
- It should be x+15＝35.
- So even though they were trying to get rid of this 2x
- and they should have subtracted 2x from both sides
- they inadvertently added 2x here
- so they said 2x+3x＝5x.
- Well, that was wrong.
- You have to subtract.
- And you're just left with x+15 ＝35.
- So step 2 is when they made their first error.
- Problem 120.
- 120
- A 120-foot-long rope is cut into 3 pieces.
- The first piece of the rope let me draw.
- This is begging for a diagram.
- All right, so they say the first piece of rope
- is twice as long as the second piece of rope.
- Ok, and then they say the third piece of rope
- is three times as long as the second piece of rope.
- Everything is a multiple of the second piece of rope.
- So the second one was the shortest one
- just reading it, right?
- The first is twice as long as the second.
- The third is three times as long.
- So if I just drew it.
- So if this is the second length of rope,
- let's call that x.
- That's the second length of rope.
- And they told us the first piece of rope
- the first piece of rope is twice as long
- as the second piece of rope.
- OK, so if this is the first piece of rope right here
- it's going to be twice as long as the second piece of rope.
- So that's 2x.
- Fair enough.
- And then they tell us
- I'll do this in magenta
- The third piece of rope is three times as long
- as the second piece of rope.
- So this is the third piece of rope here.
- It's three times as long as the second piece of rope.
- So it's 3x.
- And they say, what is the length
- of the longest piece of rope?
- All right, so let's think about it.
- If I add up all of these segments, right
- what do they add up to?
- Well, they tell us that it's 120-foot-long rope.
- So let me add that up.
- So if I do 2x plus x plus 3x
- that's going to equal 120 feet.
- That is equal to 120 feet, right
- That whole distance is 120.
- So 2x plus x is 3x plus 3x
- you get 6x is equal to 120
- You divide both sides by 6
- You get x is 120 divided by 6 is 20.
- All right, that's what x is.
- That's the second piece of rope
- or the shortest piece of rope
- because they tell us that.
- The other ones are some multiples of that length.
- But they want to know
- what is the length of the longest piece of rope?
- the longest piece of rope
- Well, the longest piece of rope is clearly piece 3.
- Right? It's three times the second piece.
- Piece 1 was only two times.
- So this is what they want to know.
- They want to know what 3x is equal to.
- So 3x is equal to three times 20
- which is equal to 60 feet.
- And that's choice C.
- Next problem.
- Let's see if I have time.
- Yeah, I think I have time.
- I might as well attempt to fit it in.
- All right.
- Don't want to rush them, though.
- Let's see.
- The cost to rent a construction crane is $750 per day
- plus $250 per hour of use.
- What is the maximum number of hours
- the crane can be used each day
- if the rental cost is not to exceed $2,500 per day?
- OK, so how much is the cost per day?
- So the cost per day is equal to $750.
- So no matter how little or much we use it
- we have to pay that $750 per day.
- So it's equal to $750 plus $250 per hour
- so plus $250 times the number of hours.
- This is what our total cost is per day.
- And they want to say
- what is the maximum number of hours
- the crane can be used each day
- if the rental cost is not to exceed $2,500 per day?
- So this whole thing has to be
- less than or equal to $2,500.
- The total cost has to be less than or equal to $2,500.
- So now let's just solve this equation right there.
- And I'll do it up here.
- So we know that $750 plus $250
- let me do it down here actually.
- More space.
- So what's the first thing we could do here?
- We could subtract 750s from both sides of this equation.
- So we get 250 times h is less than or equal to
- 2,500 minus 750, that's what?
- 1,750?
- Yeah, because 1,500 minus 750 is 750.
- 1,750.
- I just subtracted 750 from both sides of this equation.
- Let me write that down.
- So I said minus 750, plus that.
- And then I said minus 750.
- And, of course, minus 750 plus 750, that goes to 0.
- That's why you don't see anything over here.
- And on the right-hand side, 2,500 minus 750 is 1,750.
- All right, now let's just divide both sides by 250.
- We don't have to do anything to the inequality sign
- since 250 is a positive number.
- So the hours, the maximum number of hours
- the hours have to be less than or equal to 1,750 over 250
- which is-- let's see, 1,750 divided by 250.
- That's the same thing as 175 divided by 25.
- 25 goes into 175, what is that?
- I want to say seven times?
- 7 times 5 is 35.
- 7 times 2 is 14 plus 3 is 17, right.
- So h has to be less than or equal to 7 hours.
- And that is choice C.
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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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