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Contents

## Distance from zero

For this last lesson, we're going to combine everything we know about equations and inequalities, and we're going to use absolute value. >> Let's see if we can remember this from Unit 1. >> Here's negative 3 and here's positive 5. How far away are these numbers from 0? >> Put your answers in these boxes.

## Distance from zero

The distance of negative 3 to 0 is just 3 units and the distance from 0 to positive 5 is 5 units. Remember, when we want the distance from 0, we want the absolute value of a number. Absolute value is always positive. If you got that right, great work. You really remember your stuff from Unit 1. If you're a little bit fuzzy, that's okay. You can always look back at content to review what you've learned. As you progress through this course, you'll often see material that we've already covered. Math tends to build upon itself and it's no different from Algebra. Be sure to go back to previous content if you're ever unsure. Reviewing can always help you out.

## Absolute Value

We're not just going to learn about absolute value. We'll also going to learn about their equations and inequalities. We learn that the absolute value of a number represents the distance from zero. So, the distance of some unknown number must be 7. Well, I can't be 7 units away on my current number line, let's extend it. This means that variable d could be 7 units to the left of 0, or 7 units to the right of zero. The Algebra tells us that d could be at negative 7, or d could be at positive 7. This is why when we solve absolute value equations, you'll often see us split it into two equations, one with a positive answer, and one with a negative. Because remember, our distance from zero could be to the right or to the left. The negative represents the left. But keep in mind, this distance is still positive. Only the value of d or the location, is negative.

## Absolute Value Equations

We can use a similar approach for a problem like this, the absolute value or the distance of 2 d plus 1 is 7. Notice I can draw the same diagram. This could be the location of 2 d plus 1 at negative 7, or 2 d plus 1 could be positive 7. The distance of this expression is 7 on both sides of the number line. So, we're asking ourselves, what value of d would make this true? What do you think? What would d equal in this case and d equal in this case? Take some time to think about it.

## Absolute Value Equations

Well for one possibility 2 d plus one can be at negative 7, or equal to negative 7, and for the other case 2 d plus one can equal positive 7. >> You might have thought about this in your head, or maybe you did some equation solving. >> We can subtract 1 from both sides to get 2 d is equal to negative 8 then divide both sides by 2, >> So d would equal negative 4. >> For this equation, we subtract 1 from both side to get 2d equals 6, then we divide both sides by 2, so d would equal 3. >> If I think about plugging 3 in for d, I get 6 plus 1 which is 7. >> And if I think about plugging in negative 4 in for d, I would get negative 8 ... >> Plus 1, which is negative 7. >> So there are two possible values of d that make this equation true.

## Absolute Value Equations Explained

In general, when we solve an absolute value equation, we want to split it up into two parts. One with an answer that's positive to represent the distance to the right of zero, and one with an answer that's negative to represent the distance to the left of zero, or negative 7. And here, since we have multiple answers, we can list our answer as a solution set. So, the solution set is negative 4 and 3. Remember to use the curly braces for the solution set. In general, we usually list the numbers from least to greatest.

## Make It True

This question is a little bit more challenging. >> Here, the absolute value of p plus 3 and 4 has to equal 12. >> Let's think about this p plus 3 as a number. >> I know p is really a number, so this inside part will be some number. I won't know it though. >> Which numbers make this statement true? Check all the boxes that apply. >> And again, think about what the meaning of absolute value is. >> Good luck.

## Make It True

If I started with 0, I know the absolute value or the distance of 0 from 0 is, well, 0. So, 0 plus 4, that doesn't make 12, so 0 is out. If you try positive 3, the absolute value of positive 3 is just 3. So, 3 plus 4, that doesn't make 12 either. If we tried negative 3, the absolute value of negative 3 is positive 3. And I know that 3 plus 4 makes 7, which isn't 12. So, negative 3 is out, too. How about 8? Yeah, the absolute value of 8 is positive 8 and 8 plus 4 is 12. The same would be true for negative 8. If I take the absolute value of negative 8, I get positive 8 and positive 8 plus 4, that's 12. If you try negative 12 and 12, these won't work either. This is great. We know the number can equal 8 or negative 8.

## Isolate the Absolute Value

But where does this show up in my equation? If we subtract 4 from both sides, we're going to isolate the absolute value. We'll get the distance of p plus 3 alone. This is where the 8 shows up. The absolute value or the distance of p plus 3 from 0 must be 8. When solving absolute value equations, we need to make sure we get the absolute value symbol on one side of the equation alone. That way, we can represent the distance. Once we isolate the absolute value, we want to check. We want to make sure that it's greater than or equal to 0. Remember, distance is always positive, so if we had a negative answer on this side, there would be any number that could make it true, there'd be no solution. This is a difficult concept and it's one that we'll stumble upon, too, later. So now, we're back into the same case before. The value of p plus 3 could be 8 units to left of 0 at negative 8, or it could be 8 units to the right of 0, at positive 8. So, let's split this equation up into 2 equations, setting p plus 3 to negative 8 and p plus 3 equal to positive 8. Just like from before. And at this point, you might even be able to solve it in your head, which is great. For more complicated questions, it's best to stick to the equation solving. Subtracting 3 from both sides, p could equal 5. And subtracting 3 from both sides, p could equal negative 11. So, the solution set or the values for p, are negative 11 and 5. And just like all our other equations, we can check these values in our original equation to see if we are right. That's one of the beauties of mathematics. This can help build your confidence in math. If this don't check, maybe it's a good idea to look back at your work. You might rethink your process or catch a small error.

## Check the Distance

Here's another absolute value problem. >> The absolute value of 2m plus 8 is negative 4. >> So what's the solution to this equation? Choose the best answer.

## Check the Distance

If you drew out a diagram, it probably looked like this. But, the distance from zero was negative 4, and that can't be right. The absolute value of any expression, or number, is always greater than or equal to zero. So, there's no value of m that can make this true, there's no solution. If you were solving the equations like before, you might have thought it was negative 2 and negative 6. But, let's check negative 2 to see the result. I can plug negative 2 in for m. 2 times negative 2 is negative 4, and negative 4 plus 8 is positive 4. We know the absolute value of this 4 is positive 4. It's the distance from zero. And it can't be negative 4. It's not true. And that's why none of these solutions work.

## Absolute Value on Both Sides

Here's our last absolute value equation. This one looks crazy. We have the absolute value of 2x plus 1 is equal to the absolute value of 3x minus 5. We have the symbol on both sides of the equation. Alright, let's take a step back and think about what this really means. We know x is some number, so if I plug in a number here, I'll get another number. The same is true on this side. So these numbers could be positive or they could be negative. So if the number's positive and this number's positive, is this statement true? Or if there is a negative sign in front of the number, is this statement true? Check all the statements that you believe are correct. That is the left side is equal to the right side.

## Absolute Value on Both Sides

Let's look at this first case. If I choose a number like 3, I get the absolute value of 3 equal to the absolute value of 3. The distance of these numbers from zero are both 3, so that one works. If I make the second one negative 3, well, the absolute value is still 3. The distance of these numbers from zero is 3. This one is correct and this one is also correct. We just switch the sides that the absolute values are on. And finally, if we had both negative numbers, well, that would be correct as well. The distance of negative 3 from zero is 3. So, all these cases are true. We're going to use this information to figure out how to solve this problem. Let's take a closer look.

## Solve It Twice

When thinking about solving this problem, we know that the insides can both be positive numbers, so we leave them alone. We also said that they could both be negative. Remember, a negative number equaling a negative number. But this equation is the same one we just had. We've just multiplied both sides by negative 1, so I don't need to solve this. The other case we had was a positive number and a negative number. This second number or this expression could be negative, or we could have this first number be negative, and that case this first part would be negative. But notice, if I multiply both sides by a negative the same equation, I don't need to solve it. Whenever we solve the absolute value on both sides, we always write 2 equations. We keep both sides positive and then we multiply one side by a negative. And again, it doesn't matter which side has the negative, usually we make the second one the negative. Alright, to wrap up this problem, what's the solution set to this absolute value equation? Be sure to write your answer as a set.

## Practice 1

Thanks, Chris, for telling us all about how to solve absolute value equations. >> For the first practice problem I want you to try, we're going to have the absolute value of 2x minus 5 equals 3. >> You can put your two answers in these two boxes, because we know absolute value equations have two answers.

## Practice 1

Did you get x equals 1 or x equals 4? If so, well done. Let's see how we got these answers. We know when we're solving absolute value equations that we have to split it into two cases. We have the case when 2x minus 5 is positive, when it will equal 3. And we also have the case when 2x minus 5 is negative 3. So we're going to solve each one of these equations independently. When we add 5 to both sides of the equation here we get 2x equals 8. Which means x equals 4 when we divide both sides by 2 and when we solve this equation add 5 to both sides I get 2x equals 2 and when I divide both sides by 2 I get x equals 1. x equals 1 and x equals 4 are the solutions to this first absolute practice problem. Lets try a few more.

## Practice 2

For the next practice problem on solving absolute value equations, we're going to try the absolute value of 3x plus 6 equals 0. Now since 0 is neither positive nor negative, there's only going to be one solution. You can put your answer in this box. So do your best.

## Practice 2

Did you get x equals negative 2? If so, great work. Let's see how we got x equals negative 2. Since my absolute value is equal to 0, and 0 is positive nor negative, there's only going to be one case, which is 3x plus 6 equals 0. If we subtract 6 from both sides, we have 3x equals negative 6. Divide both sides by problem?

## Practice 3

Let's try an absolute value equation that looks a little different. >> Here we have the absolute value of 4x minus 7, plus 2 equals 7. >> You can put your answers here and here.

## Practice 3

The solutions to this absolute value equation are 1 half, or 3. If you got the right answers, great job. Let's see how we got these answers. This time, I have an absolute value equation, but I have an extra term next to the absolute value. Before I can split my absolute value equation into two cases, I have to isolate the aboslute value first. So I can do that by subtracting. 2 from both sides. Then, I have the absolute value of 4 x minus 7 equals 5. Now that my absolute value is isolated, I can split it into two cases. I have 4 x minus 7 equals 5, and 4 x minus 7 equals negative 5. So, let's solve this one first. I add 7 to both sides, 4 x equals 12. I divide both sides by 4, and I get x equals 3. For the negative case, when I add 7 to both sides, I get 4 x equals 2. When I divide both sides by 4, I get x equals 2 4th. Remember, whenever I have a fraction, I have to make sure I put it in lowest terms or in reduced form. So, 2 4th is the same as 1 half. And that's how we got 3 and 1 half. Let's try just a couple more absolute value equation problems.

## Practice 4

For the next question, this one looks a little different. I have the absolute value of 2x plus 3 equals negative 5. Now, I want you to think very carefully about this one before you start trying to solve it. You can put your answer here. If there's no solution you can put in ns for no solution. Good luck.

## Practice 4

This absolute value equation was tricky. >> The answer is actually, no solution. >> Why is the answer no solution? If you recall, absolute value tells us the distance from 0. >> Since one side of our equation is an absolute value, we know that absolute values are the distance from 0 and they're always positive. >> Do it make any sense to have a negative distance? ... >> No, we can not have a negative distance. >> There is no value of x that will make the absolute value of 2x plus 3 equal to negative 5. >> It's an impossible problem. So, the answer is no solution. >> Let's try one more absolute value equation.

## Practice 5

For our last question on absolute value equations, I want you to try to solve when one absolute value equals another absolute value. >> You can put your answers here. >> I know this problem is a little more difficult than the others, but do your best.

## Practice 5

That was a more difficult problem. If you got negative 3 or negative 1 5th, congratulations. I would say that you have mastered absolute value equations. Let's see how we got those answers. When I have two absolute values that are equal to each other, I still have to make two cases. And the two cases for this problem are going to be 3 x plus 2 equals, the first case is going to be exactly what's inside the absolute value, which is 2 x minus 1. The second case, we keep the left side of the equation the same and make the right-hand side of the equation negative, so I have negative 2 x minus 1. Let's go ahead and solve these. First I subtract 2 x from both sides. When I subtract 2 x from both sides, I get x plus 2 equals negative 1. And when I subtract 2 from both sides, I get x equals negative 3. So, there's one of my answers. we have to be a little more careful here. I have negative parentheses, 2 x minus 1, which means negative 1 times 2 x minus 1. Which means I need to distribute that negative sign. Negative 1 times 2 x is negative 2 x, and negative 1 times negative 1 is positive 1. From here, I'm going to add 2 x to both sides, which gives me 5 x plus 2 equals 1. When I subtract 2 from both sides, I get 5 x equals negative 1. Divide both sides by 5, and I get x equals negative 1 5th. That problem was more difficult then a lot of the absolute value equations. So, if you got the answer right, I'd say you've mastered absolute value equations. And, we're ready to move on to absolute value inequalities.

## Absolute Value Inequalities

Now that we've seen absolute value equations, let's try absolute value inequalities. >> Which of these number lines has the solution set that could satisfy this inequality? Think about what absolute value means, >> That will help you find the numbers.

## Absolute Value Inequalities

We want the absolute value of f to be greater than 5, or the distance of f from zero to be larger than 5. If f were here, it wouldn't be more than 5 units away, so this number line is out. If f were out here, beyond negative 5, then the distance to zero would definitely be more than 5. So, this is part of the solution. But there are also other values over here, like 6 and 7, which have a distance greater than 5. So, we want both of these regions. If you chose this one, nice work. Any number in either region, this one or this one, would work.

## Less Than

When we have a greater than inequality, we get a union. Notice that f could be greater than 5, or f can be less than negative 5. Just like in the absolute value equations, we have two inequalities. One with a positive answer and one with a negative answer. But for the negative answer we had to reverse the inequality sign. This will come up in problem solving. But, what about a less than sign? Here, we have the absolute value of a number is less than 5, so we want all the numbers whose distance is less than 5 from 0. So, let's have another check. What sets of numbers satisfy this less than inequality? Choose the best answer.

## Less Than

If f was at negative 6, the distance to 0 would be 6, which is greater than 5, so this number line's out. If f were at positive 6, that distance would also be bigger than 5. It would be a distance of 6, so this number line's out. This number line was a combination of these two, so we know this can't be it either. Sure enough, it's this number line. Every number in here is within 5 units of 0. So let's compare this greater than and less than, and see what we have.

## Between two Numbers

When we have a less than inequality, we perform the same steps. f could be less than positive 5, or f could be greater than negative 5. Remember, for the negative answer, we reverse the inequality sign. And here's what's interesting. The greater-than symbol creates a union, or two separate regions, where the less than sign creates an intersection between two points. With just the flip of a sign, we can exclude entire numbers or include a small region. And let's look more closely right here at these 2. We said we get an intersection, so that means f is between these 2 numbers. Let's rewrite these inequalities. Now that I've switched the spot of these 2 inequalities, I'm going to reverse the direction of this inequality. I've switched this inequality around, so negative Notice the f's are in the middle now. The f's are between my 2 numbers. We can even write this as one inequality, and, in fact when you solve any absolute value inequality with a less than symbol you can write it as one inequality with

## Practice greater than

Use what you know about absolute value and what we just covered to solve this inequality. >> Try your best. And if you're stuck, try watching the first couple seconds of the solution video. >> Be sure to pause it once you think you have an idea of what to do. >> You can write your answer in interval notation in the box. >> Be sure to use parenthesis or brackets, or a union symbol if necessary. Good luck.

## Practice greater than

We know this value inside needs to be greater than 7 or this value inside can be less than negative 7. Remember to catch that inequality sign, we have to reverse it for the negative answer. We can subtract 1 from both sides and then divide by inequality, I subtract 1 from both sides, then divide by 2, so x is less than negative 4. On a number line, we have x greater than 3, and x less than negative for my inequality. If x is any value in here or any value in here the condition is satisfied. In the interval notation we'd have negative infinity comma negative 4. We use parenthesis because we can include these values. Then the union because you want this or this region other than 3 to positive infinity again with parenthesis. If you got that one right, excellent work.

## Less than practice

Here's nearly the same exact problem except this time we have a less than symbol. I'm going to solve this inequality just like before. First, 2x plus 1 can be less than 7 here or 2x plus 1 could be greater than negative 7 here. But I don't have to solve these separately. I know 2x plus 1 is less than 7, but greater than negative 7. So it's in between these 2 numbers. Let's just write 1 inequality and solve it. First I can subtract 1 from all 3 parts. This gives me negative 8 is less than 2 x is less than 6. Now I just divide every part by 2. So x is greater than negative 4 but less than 3. It's between these values. We can see we get the numbers between negative 4 and 3. Using parentheses we can include these point because we don't have an equal to by here. This would be the answer in interval notation. In the last problem we had a greater than symbol and we got the outside portions of the number line. For the less than symbol, we get an intersection, the part in between. I hope this helps with your problem solving.

## Think Distance

So whenever we see absolute value bars, we want to think distance. Let's see if we can use that knowledge to answer this question. So what do you think? What could the values of K be? 0? Negative 3 and 3? Any real number? Or no solution? Take some time to think about it and draw some diagrams. Maybe try and place K at different parts of the number line.

## Think Distance

Let's say k was equal to negative 5. Well, I would have the absolute value of negative 5 is greater than negative 3. So, 5 is greater than negative 3, that's true. What if k was negative 1? I'd have the absolute value of negative 1, which is greater than negative 3. Well, the distance of negative 1 to zero is just positive 1. So, this is true, too. We know any number to the right of zero will be positive. And, in fact, the absolute value of any number is always greater than or equal to zero. So, this is always true, it doesn't matter what k is. Any real number would work on our number line. That was a tough question and I hope it got you thinking. Now that we've wrapped up this lesson, try practicing a few problems, and feel free to look back if you need to. Sometimes these concepts can be challenging.

## Inequalities Practice 1

Now that you've learned about absolute value inequalities, let's try some practice problems. >> The first problem I want you to try is the absolute value of x minus 1 over 4 is less than or equal to 2. >> You can put your answer in this box in interval notation.

## Inequalities Practice 1

So, how did you do on your first absolute value inequality? Did you get negative absolute value with a less than, I know that we're going to have an intersection, which is going to leave us one region. To solve this absolute value inequality, I can set it up as a compound inequality. Negative 2 is less than or equal to x minus 1 over 4, which is less than or equal to 2. So, here's our two cases. We have negative 2 is less than or equal to x minus 1 over 4 and we have x minus 1 over 4 is less than or equal to 2. We can solve this compound inequality all at one time. First thing I want to do is clear my fraction. To clear my fraction, I'm going to multiply through by the least common denominator which is 4. When I multiply through by 4, I have to multiply 4 by negative 2 which gives me negative 8. And I multiply 4 by x minus 1 over 4, which gives me x minus 1. And I have to multiply 4 times 2, which is 8. The next thing I need to do is isolate the x in the middle, so I have to add 1 to all three parts, and I get negative 7 is less than or equal to x, which is less than or equal to 9. And I can draw this on a number line. If this is 0 and this is 9 and this is negative 7, we know that x has to be greater than or equal to negative 7 and x has to be less than or equal to 9. So, I fill in the region in the middle. And that gives me the interval from negative 7 to 9 with closed brackets. Remember, we use closed brackets when we have less than or equal to or greater than or equal to. Let's try another one.

## Inequalities Practice 2

For the second practice problem, let's try the absolute value of 3 x plus 2 is greater than or equal to 5. You can put your answers here and here. I've already put the union symbol for you. Remember, if you want to type in infinity, you have to type in backslash inf. Good luck.

## Inequalities Practice 2

The solution to this absolute value in equality is the interval from negative infinity to negative 7 3rds unioned with the interval from 1 to infinity. Lets see how we got that. When I see an absolute value inequality with a greater than or equal, I know that I'm going to have a union. Which means that I have to separate this into two different cases, and connect them with an or. So, I'm going to write my two cases right here. The first case is, 3x plus 2 is greater than or equal to 5. It's exactly the same as the original problem, without the absolute value, and then I need the or. The second case is going to be the absolute value or 3x plus 2, and we have to switch the direction of the inequality. And we have to change the sign. So, the absolute value of 3x plus 2 is greater than or equal to 5. Is the same as the compound inequality 3x plus 2 is greater than or equal to 5, or 3x plus 2 is less than or equal to negative 5. Remember, when we're solving compound inequalities, we need to solve each of these inequalities independently. So, let's start with this one. I can subtract sides by 3, and I get x is greater than or equal to 1. For the second inequality, I do the same thing, by subtracting 2 from both sides, And I get 3x is less than or equal to negative 7. Divide both sides by 3 and I get x is less than or equal to negative 7 3rds. If I want to draw this on the number line, we'll put 0 right here. Here's 1. And negative 7 3rds is equal to negative 2 and square bracket and I know I want everything greater than. To graph x is less than or equal to negative 2 and 1 3rd I go about negative 2 and a 3rd and, since x is less than. Then I want to go in that directions, and that's how we get our interval notation here we have negative infinity to negative 7 3rd with a square bracket because it's less then, or equal to, and I union that with the interval from 1 to infinity again with the square bracket because it's greater than. Are equal to. Let's try one more absolute value inequality.

## Inequalities Practice 3

For our last absolute value inequality, we are going to try negative 3 times the absolute value of x minus 4 is greater than 6. Now be very careful when you do this problem, you can put your answer here. If this inequality has no solution, I want you to write ns in the box. Goodluck.

## Inequalities Practice 3

This problem was a little harder than the others. If you wrote ns or no solution, that's great. Let's see why the answer is no solution. When I'm solving an absolute value inequality, the first thing I need to do is make sure I isolate the absolute value. To isolate the absolute value, in this case, I have to divide both sides by negative 3. To isolate the absolute value, I have to divide both sides of the inequality by negative 3. When I divide the left side by negative 3, negative 3 over negative 3 is 1, so I'm simply left with the absolute of x minus 4. Now, since I divided both sides of this equation by negative 3, we have to remember to switch the direction of the inequality. And 6 divided by negative 3 is negative 2. Now, let's think about this problem. The absolute value of x minus 4 is less than negative 2. We know that absolute values are always positive so can the absolute value of x minus 4 ever be a negative number or much less, less that negative 2? No, this can never happen. Since the absolute value can never be negative, this inequality has no solution. Let's go on to Chapter 3, where we're going to learn about lines.