In the last lesson, an advertising agent told Grant that with her company's help, he can make at least $5,000 in two weeks. We said initially, that this meant that after two weeks, Grant's profit would be equal to $5,000. We dealt with a lot of equations in this course so far just like this one, but is this the best tool to use in this situation? As I see it, we have three options. We can either stick with describing this as profit equals 5000. We can come up with some other mathematical statement that would fit this situation better. Or we could just decide not to do any more math ever again. Pick which one you think is the best option.
I think that we need some other mathematical statement. An equal sign just does not cut it here. Grant is not necessarily going to make exactly $5,000 in two weeks. He might make more than that. If you picked the last answer, I might be very sad. Math is wonderful. Anyway, we'll talk more about these other kinds of mathematical statements right now.
So we need to find a way to use math to say that Grant's profit could be equal to $5,000, but it might also be more than that. Now what if I told you this sentence could be translated into math like this? Profit, some symbol we haven't seen before and then 5000. What then would you say this new symbol stands for?
We know that Grant will make at least $5,000. So his profit will either be equal to $5,000, if he makes the minimum amount he can make, or it will be greater than that. So this symbol stands for, is greater than or equal to.
Lets think a little bit more about this new mathematical statement, it doesn't show equality since this isn't an equal sign, and that means this isn't an equation so we are going to give it a new name, this is an example of an inequality, we'll see a few other types of inequalities a bit later on. Just looking at this one we have in front of us though does it tell us that Gram's profit has one particular value? How many different values for profit would make this inequality true? Do you think that no values for profit satisfy this? Think there's only one? Are there two? Or are there a lot?
The answer is a lot. If all we know about Grant's profit is that it could be $5000 or more, then profit could be $5000, or it could be $5001, or $5002.50, or $120,000. Or any other number that's bigger than or equal to 5000. This means that there are a ton of numbers. Way more than just these ones I've written down here. In fact an infinite number of numbers that profit could be equal to.
Well I, for one, have a pretty tough time trying to think about an infinite number of numbers. It would be great if we had a straightforward way to visualize this. Luckily, someone had the awesome idea to invent the number line, which is this wonderful visual device right here. A number line is basically a way of displaying all of the real numbers. We can represent whatever number or range of numbers we want to on the number line just by shading in certain positions on it. For example, since this dot is at the hash mark for negative 4, I have just plotted negative 4 on this number line. Even though I've shown the integers on this number line, we can also plot any point in between two integers, which is why all the real numbers can be shown on the number line. Notice that just like on the x-axis of a coordinate plane, numbers increase as we move to the right and decrease as we move to the left. I've labeled three points on this number line, A, B, and C. And I'd like you now to tell me what you think that each of these are equal to. What do you think the values at all three points are?
A is negative 2.5, B is 0, and C is at 3.5.
Going back to the inequality you came up with for Grant's profit, that profit is greater than or equal to 5,000, which of these 4 number lines do you think properly represents this inequality?
This one is correct. We can see that 5,000 is included in the area that's shaded. And then, the teal line shows that every value to the right of 5,000 on the number line, or that is, every real number greater than 5,000, satisfies the inequality for profit.
We talked earlier on last lesson about the fact that Grant wants to keep his budget for making nozzles under $10,000 this coming month. What inequality from these ones down here do you think best describes this sentence?
The answer is this second one right here, which we read as nozzle budget is less than 10000. We have that Grant wants to spend under $10,000, so he wants his budget to be something less than this. Not anything greater than 10000 and not even anything equal to 10000. This last answer right here, that nozzle budget is not equal to 10000.
Let's say that we have some number, n. And to the left of it on the number line is some number m. We're not exactly sure where it is but we know that it's located to the left of n on the number line. Which statement then, best describes the relationship between these two points?
The answer is, this one, which we read is m is less than n. Remember that numbers increase as we move to the right on the number line and they decrease as you move to the left.
So whether we're trying to write that one number is less than another number or that one number is greater than another number, the bigger side of this inequality sign is next to the larger number. Thinking about how we use this symbol, which of these statements is mathematically correct? There may be more than one right answer.
Negative 5 is smaller than negative 1. It's the left of negative 1 on the number line, so we should pick negative 5 is less than negative 1. However, it's also true then, that negative 1 is greater than negative 5, since negative 1 is to the right of the number line from negative 5. So these are both correct answers.
Let's talk a little bit about number line notation. There are two main ways that people use to show inequalities on number lines. One set of conventions uses dots, either open or closed, and the other set of conventions uses parenthesis, either square parenthesis or brackets, or rounded parenthesis. The closed dot and the square bracket are equivalent. And to one another or show the same inequalities. In the open circle and the rounded bracket show the same inequalities. Thinking about what you know about selecting numbers on number lines so far, which pair of inequalities do you think shows x is greater than negative 2 and which one do you think shows x is greater that negative 2?
These two number lines on the left show the inequality x is greater than or equal to negative 2. And the two on the right are the ones that show x is greater than negative 2. Remember what we said very early on the lesson, that if a number is shaded, it's included in the range of numbers that are acceptable for the solution. X is greater than or equal to 2, says that x can be greater than 2, but it can also equal 2. So 2 needs to be included in this range of numbers, which is why this circle is shaded in. When we have x is greater than negative 2, x is not allowed to equal negative 2, so the circle is not shaded on the point negative 2. The rounded parentheses is supposed to show that all of the values greater than negative 2, including everything just up until negative square bracket as including the negative 2, because it's telling us exactly where this range of values cuts off, precisely at, but including negative 2. So keep in mind for the future which kind of bracket shows inclusion and which one does not.
If instead we have that m is less than or equal to n this time, which number line shows all of the possible values that m could have. Remember a discussion of what these different notations mean.
This number line is the correct answer. We know that m is supposed to be smaller than n or equal to n. So we need to have a bracket that includes n to square 1. And also select all of the values to the left of n on the number line. Another thing to note is that the quantity we're thinking about, which is m in our case, is conveniently written to the lower right-hand hand corner of the number line. You'll continue to see this on different number lines in the future with different variable names, depending on what we're searching for.
Using what you've learned about inequalities so far, which of these mathematical statements down here are correct?
Here are our answers. You've learned that inequalities have ranges of values that are solutions to them. So that means that if a number falls in this range, we can use it in the inequality to make a true statement. Some of these may be a bit confusing though. For example, 3 is less than or equal to 7. On a number line, if we're curious about all of the numbers that are less than or equal to then draw a square bracket to include 7 in that range. We can see that 3 definitely falls in this region. It is not equal to 7, but it is less than 7. And it only needs to fit one of the criteria implied by this symbol. It either needs to be less than 7, or it needs to be equal to 7. Similarly, the statement course gives us the inequality 8 is greater than or equal to 8. And you might at first think, oh my gosh, 8 is not greater than 8. However it only needs to be greater than or equal to 8. It's equal to 8 so that's enough to satisfy this inequality. We can see the difference here then, between this symbol, the greater than or equal to, which implies inclusion of 8, versus the just greater than sign. This statement is true. Whereas this statement is not, because 8 is greater than 1 plus 7, only allows for the greater than criterion, not the equal to one. 1 plus 7 no longer fits the bill.
Here's a list of inequalities for you. And for each one, I'd like you to tell me which number it says could be greater, a or b, or neither. And then also, tell me if the inequality says that a and b could be equal to one another.
So here are your answers. Please note which of these expressions allow for a and b to be equal, and which ones don't. Know that the two inequalities that do not allow the possibility of equality between a and b, are the two strict inequalities, where inclusion is not allowed. So you would use the round parenthesis on the number line if we were to indicate these.
We've looked at a lot of different equalities and inequalities that use these symbols up here. What I would like you to do now, is translate each sentence I've written here into an equality or inequality by inserting one of these relational symbols into each of the boxes. Pick the one that you think fits each description best.
Take a second to look at the answers that I've written here. The two that I personally get most confused about are the two that end up with less than or equal to signs in the answers. Saying that Grant wants to employ no more than 15 staff means that he can employ any number of people up to 15, but then can't go any higher than that. So 15 is the highest number he can employ, and then he can also pick any number less than that. Similarly, if Grant will pay at most $5 per week, it means he can pay any amount up to $5. So this is the same sort of relation as in the other problem.
We just said that the amount of cleaner fluid that Grant needs is greater than have x is greater than 50. How would you represent this inequality on this number line? Notice that the 50 hash mark is right here, and I've put a question mark on that to indicate that you need to pick which of these symbols belongs at the 50 mark. Then also please tell me, which side of the number line we should shade in.
This means that x can be any real number to the right of 50 on the number line, since these are the numbers that are greater than 50. But this range doesn't include 50. So we need to pick the parentheses that doesn't show inclusion; this round one.
Reflecting on your recent experience with number lines, what inequality do you think that this number line right here represents? Please remember that the variable that this number line is assigning values to is located over here. Type your answer in this box.
The variable that we're dealing with here is x, and the number line shows that x is less than negative 4.
What if we had this number line instead? What inequality should we write in the box now?
Our variable is z, so let's of by writing that and z is greater than or equal to negative 1.
We saw earlier that Grant wants to employ no more than 15 staff. Now, if we let x equal the number of staff, then the inequality that we came up with before for this would say x is less than or equal to 15. Now my question is whether or not this is actually the best way to describe this situation. Think about hiring, and whether out of all the real numbers, the only restriction that we should place on x is that it's less than or equal to 15. Or, if there's another restriction you should place on this. Remember that the unit here is people. So keep that in mind. Should we leave this or should we change it?
I think we should change it. So, why? Why do I think we should change it? Well, if we look at the inequality we have right now, and we plot it on a number line, we get this. We have a square bracket at 15, and all of the numbers to the left of 15 on the number line are shaded. But notice that this includes numbers that are less than zero, negative numbers. And it doesn't really make sense, considering number of staff that has to do with people, to have a negative number of people hired.
Since we know that the other number line just isn't cutting it, which of these four do you think would be a better choice?
This last one is the best choice. We still want to cap the number of people that Grant can hire at 15, still including 15. But we also want to make sure that he doesn't hire a negative number of people. So we put the end of our range at 0, but including 0, because technically, he wouldn't have to hire anyone.
Now that we've selected this number line, what does this actually tell us about the value of x? We know that it has to have something to do with 0 and something to do with 15. But what exactly? I'd like you to pick one condition from the left column and from the right column that you think fits x. And then also whether both of these conditions are true, in which case you should pick and. Or, if only one of them might be true at a time, pick or.
If we only look at the area surrounding 0, we see that we've shaded to the right of 0 and also included that point. So, that means x is greater than or equal to 0. Now, if we only look at the area surrounding 15, 15 is included and then, we shade to the left. So, x is less than or equal to 15. Every number within this range that's shaded is both greater than or equal to 0 and less than or equal to 15. So we pick and.
Right now, we have two separate statements about the values that x is allowed to equal. But looking at our number line, there's only one area of the number line shaded in. [unknown] then, it would seem that we could write this as a sort of connected inequality, combining these two inequalities together. So, how would you guess that we could write this as, what I'm going to call, a chained inequality? Insert the inequality symbols that you think belong in this two boxes.
We can write 0 is less than or equal to x, which is less than or equal to 15. X is greater than or equal to 0 can be written in, in a different way, as 0 is less than or equal to x. Now if we put this side by side with our other inequality and we match up the x's in the same spot, we end up with this compound inequality.
Here's another number line for you to look at. How shall we express the values of x shown here, as a compound inequality in chained notation like we saw on the last example?
The answer is negative 2 is less than or equal to x which is less than 6. The negative 2, with its square bracket, is included in the range, but the 6, because of the round parentheses, is not.
There's another kind of notation that we can use to describe the fact that x can equal anything within this interval of values. Thinking about our number line here, which of these four representations of this interval do you think fits our situation?
Notice that the two symbols to enclose the ranges of values listed down here are square brackets and round parentheses, just like we see on our number line. Since up here, we have a square bracket to include the negative 2 in our range, and we have a round parentheses to not include the 6, we'll just match those with either end of the interval that they belong to down here. So this is the correct answer.
This kind of notation that we just used is called interval notation.
Here's another number line for you. How would you represent this inequality, shown by the number line, using interval notation?
We know that the values here lie between 4 and 8, so I'll start by writing those numbers. The 4 is not included in the range, and neither is the 8. So, this is our answer.
In order to use interval notation to write the possible values that x can have, we need to figure out the lower bound and the upper bound of this interval. I think the lower bound will be too difficult in this one, but I'm curious about what maximum value you think x can take on. Please pick. Do you think it's 0, 5, 10, or, do you think there isn't one?
There isn't one. We can see that we have an arrow pointing to the right side of the number line, in the filled-in section for x, which means that the set of values that x can equal extends forever in this direction, in the positive direction. X can equal any number greater than 5. There's no upper limit.
Now there you thought a little bit about what x can equal. Can you write the set of possible values of x using interval notation? Please type in the parentheses or brackets that we need in the tail boxes and then for each light blue box, please select one of the values down here, for either one, as the number that belongs there.
The lower limit of this inequality is 5, but we can see that it's not included in the interval, so we need a round parenthesis. We said that there is no maximum value that x could take on. That means that the possible values of x could extend to infinity. And, whenever we have infinity in an interval, we need to use a round parenthesis. Infinity isn't a number that we can have a hard stop at. It can't be included in an interval, because it's bigger than anything we can quantify. So, we have to use a round parenthesis. If you had trouble getting this, no big deal. I know this is a pretty new concept and definitely a new notation with the infinity and how to use it with parenthesis. But, now you know.
Here's another number line for you to look at. And this time, I'd like you to type in the entire interval of values that y could equal. Don't forget to use your proper notation.
Since our shaded area of values for y ends in an arrow pointing to the left, the lower limit of this interval is negative infinity. There is no lower limit, in other words. We can see that the interval ends at 30, with a square bracket, so or equal to 30, which means that y can have any value that's either 30 or something lower than 30. And remember, we have to use this round parentheses next to our infinity, even it's a negative one.
Early on, we saw that if we have a compound inequality that's written in a chained form like this, it's really the same as having two inequalities joined together with an and. We know that with this inequality, x needs to lie between negative 2 and 6, where 2 is included in the interval. So it's true that both x is greater than or equal to negative 2, then we have x is less than 6. Let's graph each of these individually on a number line, and then, the compound inequality on a third one. First we have that x is greater than or equal to negative 2, or in other words, that negative 2 is less than or equal to x. Then we have, x is less than 6. And then if we graph these two together, what we're really doing is finding the shaded part of the number line between the two that overlaps. So, if for example, I lay this number line on top of this number line, and I then shade in the part that's overlapping, I get exactly what we have in the compound inequality down here, because of the and. We have a double restriction happening, and that further reins in the values that x can equal to this smaller range than we would have if we had either one of these true on its own.
So when we see the word and, it means that both of these inequalities must be satisfied for the value of x. What I'm curious about is what changing this word to or does to the relationship between these two inequalities and how that's going to affect our number line, or our range of values that x is allowed to equal? Does it mean that neither of these inequalities needs to be satisfied? Does it mean that only one of them at a time is allowed to be satisfied, not both? Does it mean that at least one of them needs to be satisfied, but that both could be? Or does it mean that both of these inequalities need to be satisfied?
The word or means that either this inequality or this inequality needs to be satisfied. But it doesn't rule out the possibility that both of them could be satisfied. So, the third answer is the best choice.
Now that you have an idea about the difference between what and, and or denote with inequalities. I'd like you to match each of these number lines on the left with the compound inequality that it represents. These are definitely a little bit tricky. And I know that or is kind of a new concept for us. But just give it a try. Start with the ones that you're sure about. And then move onto the ones that you're not totally positive about.
This first inequality here, negative 2 is less than or equal to x and x is less than 6 is choice c. This is the one we already talked about. We know that the area shared by the two inequalities here is the one that needs to be shaded in, which is just this area. Now let's try an or. Negative 2 is less than x, or x is less than 6. This is definitely tricky. The way that I like to try to graph these, is to start by putting the proper parentheses or brackets at the proper places, and facing the proper direction. So, first we have negative 2 is less than or equal to x. I come down to negative 2, and I know this needs to be a square bracket because of the less than or equal to. And I know that values of x are going to lie to the right of negative 2 and including negative 2, so I put a square bracket facing to the right. Now, we have this or x is less than 6. So I come over to 6, and I put a round bracket facing to the left, since x is less than 6. Now, or means that all acceptable values of x will either fulfill this criteria, or this criteria. So that means that, they'll either be in this range, anything greater than or equal to 2, or they'll either be in this range. And of course anything in the middle is fine, too. So this is the entire number line shaded in. That's choice A. Probably not what we would've expected when we just looked at this. You can see what a huge difference there is when we switched from and to or. Nagative 2 is less than or equal to x and x is greater than 6. Start with a square bracket pointing to the right at negative 2. And this area needs to overlap with some area that's graphed by the x is greater than 6 line. We know that if we graph this first inequality, it's going to be everything greater than or equal to negative 2. And the second inequality is going to share everything that's shaded by that line that's greater than 6. So, the only thing we actually need to shade is this section, the greater than 6 part. So that's choice D. And, finally, our last one. Well, we know the answer must be B, because that's the only one that's left, but let's graph it anyway. Negative 2 is less than or equal to x. So as before, square bracket at negative 2 pointing to the right. Or, x is greater than 6, so that give us this. That means that we need all of these values that are greater than or equal to 2 to be filled in, because we need to include all the values that fit either of these things. Everything that's greater than 6 is also greater than or equal to negative 2. So the number line for this looks exactly like the number line for this inequality alone would look. That's choice B.
Now that we have number lines for all of our compound inequalities, can you write these inequalities using interval notation?
The range of x for this first inequality goes from negative infinity to positive infinity. Remember, of course, that we need the round parentheses, not including the infinities. The second goes from negative 2 including negative 2 up to infinity. The third starting at negative 2, including it, up to 6, not including it. And the last one starting at negative 6 but not including it and up through infinity.
Let's say we have a compound inequality x is less than 3 or x is greater than 7. How do you show this on a number line? Please fill in what parenthesis or brackets you think belong in either of these two light blue boxes. And then also check off which of these three regions you think should be shaded in. I'll assume that if you shade in either of these two regions on the ends, you'll want the shading to extend forever in this direction or in this direction. Depending on which end you shade. If you shade in the middle however, I'll assume it's only between 3 and 7.
We would show x is less than 3 like this. And if we were to graph x is greater than 7, we would get this. Since we have an or, we know that we need to have both of these regions here. So you should have checked this and this and put two round parentheses facing outward.
You notice that this number line looks pretty different from other number lines we've seen before. And that there are two separate regions that are shaded in, There's not one region that's all connected but two disconnected regions. How then will we show this in interval notation?
Well, we know that if we only look at x as less than 3, this would be the interval we would have. And, if we were only to look at x as greater than 7, we would have this. Remember that interval notation shows us what the minimum and maximum values that x is limited to are. And, of course, whether or not those are included are shown by the brackets. But, in this case, we need to include both of these regions. We do this by showing a u between the two of them. This u here, stands for union. All the values that it can take on are shown by having a union of this region with this region. We're taking all these numbers from both areas and putting them together in one big heap. X will be somewhere in that one big group of numbers.
Here are two number lines for you, and for each them I'd like you to write the solution in both interval notation and as compound inequalities.
Let's start with this first number line. The first shaded region down here starts all the way out at negative infinity, and comes up through negative 9, including negative 9. So we have negative infinity to negative 9 united with negative 6, which is not included up through positive infinity. Written as a compound inequality, this should be read as y is less than or equal to 9 or y is greater than negative 6. Of course, noting that our variable here is y as shown at the edge of the number line. For the second number line, our interval goes from negative 4, including negative 4, through 8, which is not included. The inequality for this would be read as negative 4 is less than or equal to z is less than 8. We could also write this of course, using an and. Negative 4 is less than or equal to z and z is less than 8.
As much as we've talked about Grant, we haven't talked about his actual product that much. I think it's time that we discuss some of the specifics of wiper blade engineering. Now in general, Grant aims to have his wiper blades be exactly 2.5 centimeters long. Please note that this drawing is not to scale. However, in the manufacturing process things aren't perfect. To acknowledge that, Grant has created a tolerance range that wiper length is allowed to meet. As long as the wiper is within 3 millimeters of this total length, either 3 millimeters longer or 3 millimeters shorter, then it's acceptable and can be put on the market. Let's call the blade length in centimeters of a given wiper x. What we're really concerned with then, is the difference between x, the blade length, and 2.5 centimeters, the target blade length. If x is longer than 2.5 centimeters, then you want the difference between x and 2.5 centimeters to be less than or equal to 0.3 centimeters, which is the same as 3 millimeters. However if x is shorter than 2.5 centimeters, then you want the difference between x and 2.5 to be greater than or equal to negative 0.3. Since x could be something like 2.4, if it's shorter, and 2.4 minus 2.5 is negative 0.1, which is greater than or equal to negative point 3. Taking into account all the possible values of x, we want both of these things to be true. So, we need to insert an and between these two values. If we were to show this range of values for x minus 2.5 on a number line, you would get this. The region is, just like shown in our inequalities, symmetric about zero.
Looking at the compound inequality we have, which we can also write in interval notation, of course. Seems like, considering we have the same number, almost, written twice, just one of it with a negative sign. There should be some sort of simpler way to write this. It's not exactly redundant right now, but it seems a little bit redundant. What we really know is that we want the difference between either end of the tolerance is just .3, either in the positive direction or the negative direction. One way we can denote this is by using what's called absolute value signs. We can think of the absolute value of a number as the distance between that number and 0. This distance is absolute in that it ignores direction. We can see that counting from positive point 3 back to zero takes us point 1, point 2, point 3. But the same is true of negative point 3. Point 1, point 2, point 3. So positive point 3 and negative point 3 have the same absolute value, because they're the same distance from zero on the number line. Now that you have an idea of what absolute value is, can you tell me what the absolute value of negative 9 is? And what the absolute value of 4 and a half is? The number line is here just for you to have a visual in case you feel like counting on something.
The absolute value of negative 9 is just positive 9. If we mark negative 9 on the number line and we count back to 0, you'd have to count 9 spaces. The absolute value of 4 and a half is just 4 and a half.
Let's say that we have some real number a, that is nonnegative. As in, a is just not negative. It's either zero or it's positive. What then can we say the absolute value of a is equal to? And if make a negative, if we multiply a by negative 1, then what is the absolute value of that number?
Absolute value of positive a is just the distance from that to 0, a minus 0is just a. But the same is true of negative a. It takes just as much distance to get from negative a to 0, as it did from a to 0. So that's the absolute value of negative a is a. So if a number is positive, it's absolute value is just equal to itself. And if a number is negative, we get the absolute value by multiplying that number by negative 1.
Absolute value signs are sort of like a type of parenthesis, in that when we use them with expressions, we simplify the part of the expression that's inside the absolute value signs, before we deal with our entire quantity in relation to other things. So, I'd like you to find the absolute value of each of these quantities, and when you do that, first simply what's inside the absolute value signs, and then take the absolute value of that number.
So, our answers here are 7, 7, 17, and 17. Note that, as I suggested, I simplify what was inside the absolute value as my first step. So for example, here, I said, first off, 12 minus 5 is 7, rewrite the absolute value signs around that and the absolute value of 7 is 7. Please note that none of these answers are negative. These four numbers at least are all positive numbers. Keep that in mind as we move forward.
Thinking about what you've learned so far, what do you think the absolute value of 0 is?
The absolute value of 0 is just 0. On a number line 0 is right here and the distance it takes to count from 0 to 0 is 0. So the answer is 0.
Like I mentioned before, it's often useful to think about absolute value in terms of distance on a number line. Here, I've marked the points 5 and 14. So, if I want to count from 5 up to 14, it's going to take me 9 spots. The same is true if I want to count from 14 back to 5. It's going to take me 9 places along the number line to get there. So once you get from negative 12 to 5, I can see with my number line that it is going to take me. I'll spare you counting, 17 steps. Considering we know that this distance is 17, which of these following absolute value expressions shows that distance?
The answers are, the absolute value of negative 12 minus 5 and the absolute value of 5 plus 12. You can see that these are both equal to 17, although this one takes the absolute value of negative 17, and this one takes the absolute value of positive 17.
So, we do still have the distance between negative 12 and 5, can be shown either by taking the absolute value of negative 12 minus 5, or by taking the absolute value of 5 plus 12. You'd think, though, that there would be a consistent method for finding what goes inside the absolute value signs, or finding the distance between two points. And, if fact there is. I'd like to remind you that 5 plus 12 is actually equal to 5 minus negative 12. So I'm going to rewrite this absolute value expression in this way. So considering this idea what would you say that the absolute value of 14 plus 7 is the distance between? What are the two points that this measures the distance between?
The absolute value of 14 plus 7 measures the distance between 14 and negative could find this two ways. So we either have negative 12 minus 5, or if we go the other way, we have 5 minus negative 12. 14 plus 7 is the same as 14 minus negative 7. So the other way that we could write this distance will be by starting with negative 7 and then subtracting 14. Of course both of these gives the distance of 21, between 14 and negative 7.
Let's look at an absolute value problem with an unknown in it. In fact, an unknown and an inequality. How do you translate this inequality that's written mathematically right now, into English? The distance between what and what is either less or greater than something.
This says that the distance between x and 4 is less than 5. Of course, we know that you also could have the distance between 4 and x is less than 5, since the absolute value of x minus 4 and the absolute value of 4 minus x are equivalent to one another. They both measure the distance between these two points.
We just said that we can translate this inequality as the distance between x and visualize where you think x is allowed to lie. Considering what we know about how far it can be from 4. Then write down the interval form of the solution, and also the inequality form of the solutions for, for x. So again, first just figure out the range of solution on the number line that you think x is allowed to equal, and then just translate this into the 2 types of notation we know how to use.
If the distance between x and 4 needs to be less than 5, then we can either count up 5 from 4, which takes us to 9, and draw a round parenthesis there. Or we can count down from 4 and 5, which takes us to negative 1, and we can write a round parenthesis there as well. Any area between these two points is fine as a value of x. Looking at this interval, this is something we definitely know how to handle. These solutions go from negative 1 to 9, non inclusive. And we can also write this as negative 1 is less than x is less than 9.
Before we said that we can translate our absolute value inequality, to say that the distance between x and 4 was at most 5. Now, however, we have the distance between x and 4 is at least 5. How would you translate this into a mathematical expression? Just as a hint try to use some absolute value signs and an inequality.
Right off the bat we know we have to deal with the distance between x and 4, so I can write the absolute value of x minus 4. And if this is at least 5, then I can translate that as, that this is greater than or equal to 5. So that's our answer.
Now that we've come up with this new equality. The absolute value of x minus 4 is greater than or equal to 4. How should we show that on a number line? I've already selected three points for you where we may or may not need a kind of parenthesis or bracket. Those points are negative 1, 4, and 9. And, of course, those were all important when we were dealing with the absolute value of x minus which symbol belongs on the number line at that point. You'll notice I've also drawn four boxes down here for you. One for the area of the number line to the left of negative 1. One for the area between negative 1 and 4. One for the area between 4 and 9 and one for the area for the right of 9. I'd like you to check off which of these areas of the number line should be shaded in.
If we know that we want the distance between x and 4 to be greater than or equal to 5, then we just start at 4 and count over 5 in one direction, and then shade in everything beyond that. So we get to negative 1, want to shade in everything to the left of that, since we want to be further away from 4. And we also want to include this point, because we have a greater than or equal to sign. And do the same thing in the other direction. Get up to 9. We know we want to go beyond that, but then also of course, include it. So, here is what you should have marked.
Let's try one that's maybe a little bit trickier. What if we have the absolute value of 2x minus 4 is less than 6? First, let's start by treating 2x as a sort of package. Can you tell me what values it lies between? Also, please tell me what interval 2x is in on the number line?
We know that we want the distance between 2x, wherever that lies on the number line, and 4 to be less than 6. So we can start at 4 and count over 6 in one direction. Let's mark that. And then over 6 in the other direction. So we know where we'll need to put symbols now. We want 2x to be closer to 4, then 6 units away. So these are the upper limits of the range it can fall in, and we need to shade in the area in between them. Neither end point is included because we just have a less than sign, not a less than or equal to sign. So these two values are negative 2 and 10. Written as an interval, we have this.
Now let's talk about x instead of 2x. Thinking about what answers you got in the last quiz for the interval that 2x lies in, what do you think the interval x will lie in will be? Just as a hint, if you want to check your solution on a number line, when you plug in your values of x to 2x minus 4, the opposite value of that should still be less than 6. So, be sure to check your answers.
Before, we had negative 2 is less than 2x is less than 10. To get x by itself, then, we just need to divide this quantity by 2. Since we want this to stay in the same relationship to these other two quantities as it is right now. We need to do the same thing to those quantities as well and we end with negative 1 is less than x is less than 5. This, of course, translates into the interval negative 1, 5. And, we can also plot it on our number line. To check, let's pick any random value in this region. Let's say, maybe a 0, since that's readily available. Let's see. The absolute value of 2 times 0 minus 4 is equal to 0 minus 4. The absolute value of negative 4 is just 4 and 4 is less than 6. So, that value works.