Learn about CA Algebra I: Number Properties and Absolute Value Video

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Learn about CA Algebra I: Number Properties and Absolute Value

We’re now going to take the California Standards Test Algebra I Reviews Questions. The last years I’ve been the Algebra II on. I guess I’m going on a reverse order. Let me copy and paste this first question because I think it’s good to see the whole thing. Let me see. I have copied it and I got to move this point through all the way up and then there we go. Alright, and they are asking us “Is the equation 3(2x-4)=-18 equivalent to 6x-12=18?” So let’s think about this. If we just distribute this 3, what do we get, 3x2x is 6x, 3x-4 is -12 and that of course is equal to -18. So sure they’re the same thing if you just distribute the three over the 2x-4 you get 6x-12. So the answer is definitely yes. There should be no about here and it says yes equivalents. “Yes, the equations are equivalent by the associative, no, communicative, no. “The equations are equivalent by the distributive property of multiplication over addition, right, that’s that. We distribute this 3 over the 2x-4 and they say over addition because you could use this as a plus -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 two. They say “The square root of 16 plus the cube root of eight is equal to also square root of 16. When you just have a square root there you must solve a maybe plus or minus four. When you write this way means the principle root, so it’s just plus four. You have written a plus or minus out in front if they wanted you to get the negative square root. So it’s four plus. Now, what do the third power is equal to eight? Well, two to the third power is equal to eight, right? So we could write 23=8, that’s the same thing as saying up cube root of eight is equal to two. You could also view this as 81/3. Anyway, the cube root of eight is a two, so 4+2=6 and that is choice B. Problem three, and they want to know which expressions is equivalent to x6x2?” So x6 x x2 with the same base when you're multiplying both the equations you can add the exponents, so that is equal to x to the six plus two is eight. That’s not one of the choices here. So we have to say which of this also are the same thing as x8 and so which two exponents when I add them is equal to eight, four plus three is equal to seven, five plus three and this is equal to x8 as well. So that is choice B Next problem, problem four, they want to know which number does not have a reciprocal. So the reciprocal of -1, this is 1/-1 which is equal to -1. The reciprocal of zero, that’s what, 1/0 which is not defined. So the choice is B, zero. We do not k now at one over zero is, maybe that’s a project for you to think of what it should mean and of course these have reciprocals, 1/1000 which is equal to one times a thousand of one which is equal to a 1000, and the reciprocal of three is of course 1/3. Next problem, there’s a lot of terminology here but I could say it’s good. They want to know what is a multiplicative inverse of 1/2? So essentially where can I multiply 1/2 by and then get one, right? It is the same thing as saying what’s the inverse of 1/2. So if I multiply 1/2 by the inverse of 1/2, so one over 1/2, that is the same thing as 1 x 2/1 which is equal to two. Well, another way, think over this 2 x 1/2 is equal to one. So the multiplicative inverse of 1/2 is just two and that’s choice D. Problem six, what is the solution for this equation? Sometimes this absolute value signs can appear daunting but you just have to think it through logically. If the |2x-3|=5, the total is what? That means that 2x-3=5, right? Inside the absolute value is equal to five then the absolute value of five is equal to five. So that’s fair enough. But what could 2x-3 also be equal to? What happens if 2x-3 within the absolute value sign is equal to -5? Where then you would take the absolute value of that and then you’ll get five, right? So 2x-3 could also be equal to -5. When you have seen the absolute value sign, you say “okay whatever is inside the absolute value is either five or negative five because we think the absolute value of it to get five”. So we just solve both of these equations, if you add three to both sides to this one, you get 2x=8, x=4. On the second one, you add three to both sides, you get 2x=-5+3 is -2, x=-2/2 is negative one. So x could be equal to four or x could be equal to negative one and that is choice C, x is negative one or x is equal to four. Next problem, the Algebra I once go faster than the Algebra II problems turns to be harder. Let me clear all of these. I’ll just write this one down. They say what is the solution set for the inequality 5-|x+4|<=-3. So at first this is really daunting, I cannot even make do that a logic that I did last time because of that five out there. Well, let’s think of this way. Let’s try to simplify it. You know just the absolute value of something is less than or equal to something else. So one thing we could is if we want to get rid of this five. Remember what we do to both sides of an equation or inequality whatever do to one side of an equation or an inequality, we do to both sides. So let’s subtract five from both sides to this equation, right? If you subtract five from the left side, this five disappears, right? So -5+5=0, so I’m just left with -|x+4|<= --now what’s -3-5=-8. Alright now this next step, this is something maybe wasn’t obvious for you and it make thing the in inequality there. You know if this was an inequality you would just say, “Okay, I’m going to multiply or divide both sides by negative one 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 true then if I'm multiplying both sides of this by negative one, so -1 x-|x+4|, I’m going to switch the inequalities so that is going to become greater than or equal to negative eight and I did them negative one on this side, so I have to multiply times negative one on that side and so this negative cancel out that negatives, so we’re just left with x+4>=-8*1=8. Now we can just use the logic that we had from that last problem. This tells us that the magnitude of x+4 is greater or equal to eight so magnitude. Let me draw a number line here because I really want you get the intuition of what magnitude means. So if that’s the number line and you can view a magnitude is kind of the distance from or the absolute values you can have from the distance from zero, right? So if this is zero right here and this is +8, and this is -8, the absolute value of whatever these quantities is greater than eight. That means its distance from zero has to be greater than eight. You could just say distance of zero from zero of this number has to be greater than eight. So that means at this number, greater than or equal to eight, that means at this number is definitely going to be greater than or equal to positive eight. On the number line, it would be all of those numbers, right? Or whenever we seen the magnitude so I do not care about the direction. The magnitude has to be greater than positive eight. So it also includes the negative numbers in less than negative eight and why does that makes sense? We take negative nine. We’ll see absolute value of negative nine. The absolute value of negative nine is greater than eight because nine is greater than eight. So any number to the left of negative eight or to the right of positive eight so what does that tells about this equation? So that means that--well the easiest one is x+4 could be greater than or equal to eight. So let us write that down, let me write it here, x+4>=8 and that takes in to consideration that the magnitude is greater than or equal to eight there, or x+4<=-8, that’s the magnitude to the left of this negative eight right there and now we solved it. 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 zero, magnitude of the distance from zero is say “oh, the distance from zero has to be greater than or equal to eight”. So that means this thing has to be less than or equal to -8 or has to be greater than or equal to +8. So let’s solve, x+4 is greater than or equal to eight, subtract four from both sides, you get x is greater than or equal to four. I’ll just subtract four from both sides. Subtract four from both sides here, you get x is less than or equal to -12. So the solution here is x>=4 or x<=-12 and that is choice D. Anyway, I’ll see you in the next video.

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We’re now going to take the California Standards Test Algebra I Reviews Questions. The last years I’ve been the Algebra II on. I guess I’m going on a reverse order. Let me copy and paste this first question because I think it’s good to see the whole thing. Let me see. I have copied it and I got to move this point through all the way up and then there we go.

Alright, and they are asking us “Is the equation 3(2x-4)=-18 equivalent to 6x-12=18?” So let’s think about this. If we just distribute this 3, what do we get, 3x2x is 6x, 3x-4 is -12 and that of course is equal to -18. So sure they’re the same thing if you just distribute the three over the 2x-4 you get 6x-12.

So the answer is definitely yes. There should be no about here and it says yes equivalents. “Yes, the equations are equivalent by the associative, no, communicative, no. “The equations are equivalent by the distributive property of multiplication over addition, right, that’s that. We distribute this 3 over the 2x-4 and they say over addition because you could use this as a plus -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 two. They say “The square root of 16 plus the cube root of eight is equal to also square root of 16. When you just have a square root there you must solve a maybe plus or minus four. When you write this way means the principle root, so it’s just plus four. You have written a plus or minus out in front if they wanted you to get the negative square root. So it’s four plus.

Now, what do the third power is equal to eight? Well, two to the third power is equal to eight, right? So we could write 23=8, that’s the same thing as saying up cube root of eight is equal to two. You could also view this as 81/3. Anyway, the cube root of eight is a two, so 4+2=6 and that is choice B.

Problem three, and they want to know which expressions is equivalent to x6x2?” So x6 x x2 with the same base when you're multiplying both the equations you can add the exponents, so that is equal to x to the six plus two is eight. That’s not one of the choices here. So we have to say which of this also are the same thing as x8 and so which two exponents when I add them is equal to eight, four plus three is equal to seven, five plus three and this is equal to x8 as well. So that is choice B

Next problem, problem four, they want to know which number does not have a reciprocal. So the reciprocal of -1, this is 1/-1 which is equal to -1. The reciprocal of zero, that’s what, 1/0 which is not defined. So the choice is B, zero. We do not k now at one over zero is, maybe that’s a project for you to think of what it should mean and of course these have reciprocals, 1/1000 which is equal to one times a thousand of one which is equal to a 1000, and the reciprocal of three is of course 1/3.

Next problem, there’s a lot of terminology here but I could say it’s good. They want to know what is a multiplicative inverse of 1/2? So essentially where can I multiply 1/2 by and then get one, right? It is the same thing as saying what’s the inverse of 1/2. So if I multiply 1/2 by the inverse of 1/2, so one over 1/2, that is the same thing as 1 x 2/1 which is equal to two. Well, another way, think over this 2 x 1/2 is equal to one. So the multiplicative inverse of 1/2 is just two and that’s choice D.

Problem six, what is the solution for this equation? Sometimes this absolute value signs can appear daunting but you just have to think it through logically. If the |2x-3|=5, the total is what? That means that 2x-3=5, right? Inside the absolute value is equal to five then the absolute value of five is equal to five. So that’s fair enough.

But what could 2x-3 also be equal to? What happens if 2x-3 within the absolute value sign is equal to -5? Where then you would take the absolute value of that and then you’ll get five, right? So 2x-3 could also be equal to -5. When you have seen the absolute value sign, you say “okay whatever is inside the absolute value is either five or negative five because we think the absolute value of it to get five”. So we just solve both of these equations, if you add three to both sides to this one, you get 2x=8, x=4. On the second one, you add three to both sides, you get 2x=-5+3 is -2, x=-2/2 is negative one. So x could be equal to four or x could be equal to negative one and that is choice C, x is negative one or x is equal to four.

Next problem, the Algebra I once go faster than the Algebra II problems turns to be harder. Let me clear all of these. I’ll just write this one down. They say what is the solution set for the inequality 5-|x+4|<=-3. So at first this is really daunting, I cannot even make do that a logic that I did last time because of that five out there. Well, let’s think of this way. Let’s try to simplify it. You know just the absolute value of something is less than or equal to something else.

So one thing we could is if we want to get rid of this five. Remember what we do to both sides of an equation or inequality whatever do to one side of an equation or an inequality, we do to both sides. So let’s subtract five from both sides to this equation, right? If you subtract five from the left side, this five disappears, right? So -5+5=0, so I’m just left with -|x+4|<= --now what’s -3-5=-8.

Alright now this next step, this is something maybe wasn’t obvious for you and it make thing the in inequality there. You know if this was an inequality you would just say, “Okay, I’m going to multiply or divide both sides by negative one 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 true then if I'm multiplying both sides of this by negative one, so -1 x-|x+4|, I’m going to switch the inequalities so that is going to become greater than or equal to negative eight and I did them negative one on this side, so I have to multiply times negative one on that side and so this negative cancel out that negatives, so we’re just left with x+4>=-8*1=8. Now we can just use the logic that we had from that last problem. This tells us that the magnitude of x+4 is greater or equal to eight so magnitude. Let me draw a number line here because I really want you get the intuition of what magnitude means.

So if that’s the number line and you can view a magnitude is kind of the distance from or the absolute values you can have from the distance from zero, right? So if this is zero right here and this is +8, and this is -8, the absolute value of whatever these quantities is greater than eight. That means its distance from zero has to be greater than eight. You could just say distance of zero from zero of this number has to be greater than eight. So that means at this number, greater than or equal to eight, that means at this number is definitely going to be greater than or equal to positive eight. On the number line, it would be all of those numbers, right? Or whenever we seen the magnitude so I do not care about the direction. The magnitude has to be greater than positive eight. So it also includes the negative numbers in less than negative eight and why does that makes sense?

We take negative nine. We’ll see absolute value of negative nine. The absolute value of negative nine is greater than eight because nine is greater than eight. So any number to the left of negative eight or to the right of positive eight so what does that tells about this equation? So that means that--well the easiest one is x+4 could be greater than or equal to eight. So let us write that down, let me write it here, x+4>=8 and that takes in to consideration that the magnitude is greater than or equal to eight there, or x+4<=-8, that’s the magnitude to the left of this negative eight right there and now we solved it.

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 zero, magnitude of the distance from zero is say “oh, the distance from zero has to be greater than or equal to eight”. So that means this thing has to be less than or equal to -8 or has to be greater than or equal to +8.

So let’s solve, x+4 is greater than or equal to eight, subtract four from both sides, you get x is greater than or equal to four. I’ll just subtract four from both sides. Subtract four from both sides here, you get x is less than or equal to -12. So the solution here is x>=4 or x<=-12 and that is choice D. Anyway, I’ll see you in the next video.

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