[Music] hello let's do some interval notation this is the easiest notation for mathematicians to use to write sets so if we're given an inequality then we can write it in set notation where we write it as X such that a is less than or equal to X is less than b not less than or equal to just less than or the interval from A to B so how that appears on a number line is so we're going to have I'm going to put all three number lines here so there's our three number lines so if we're looking at Just Between A and B then we're going to just put parentheses because we have parentheses for our interval because we're not equal to and then we just scribble in between the interval or we darken it or shade it in some way so that we know that that's the region we're talking about if we look that X is greater than or equal to B then what we're talking about is bracket because we're equal to and we go from B all the way to positive Infinity so we have a bracket and then we're going all the way out to positive Infinity if we're looking at the set of all real numbers for math we use a bolded r and so the way that we would write that for a mathematician the way a mathematician writes that if we're writing it in a book is we do two lines and an r and that's what we called a bolded R in mathematics and that's negative Infinity to positive Infinity with parentheses around those now notice that we use parentheses because we can never equal Infinity and when we draw it on a number line we're basically just scribbling in the whole number nine now also notice that all of these students misunderstand and think they're points but we're talking about sets not points so if I gave you this inequality and I said write this in interval notation how would you go about doing that the first thing that I do is I rewrite this as a less than sign because we want to see when we write interval notation or when we look at this on a number line we look at it as we read it and greater than's are not what we read we usually read less than so we have 8 is less than x x is the bigger value so then we're going to draw our number line and we're going to have 8 and then we're going to show the bigger part of our number line because that's the bigger region now notice I don't have a ton of tick marks I just have one drawn where the 8 is not here's zero here's all my little tick marks to get to eight I just have eight and now I'm just going to shade my region and there you go notice I use a parenthesis because I'm not equal to okay and then we have our region and we're writing our interval notation so we have parentheses 8 comma Infinity close parentheses because we're not equal to Infinity what about this region where we have X less than 4 and X is less than or equal to negative 2. so first because these are all less than we want to just draw the number line so we have negative 2 and 4. we don't have to put 0 and tick marks or any of that but we know that negative 2 is to the left of 4 and we're good now we want to shade the region we know that 4 has a parenthesis and we know that X is less than 4 and we know that negative 2 has a brace and we know that X is less than negative 2. so we're going to put the brace and the parentheses and we're going to shade now because we have an and we want the overlap of the two regions so we have parentheses 4 down to negative Infinity we have brace down to negative Infinity at two we want the overlap so that's going to give us the brace down to negative Infinity so writing the interval notation we have negative Infinity to negative 2 brace with parentheses at negative Infinity and that's our answer write the inequality in interval notation and graph x is greater than four or X is less than or equal to negative 2. now in this case we have a greater than so we want to rewrite this as a less than so we're going to have X is less than 4 so we want to write so we want to write this as 4 is less than x so that we know that this region 4 is the smaller of the two then what we're going to do is we're going to draw a number line and shade our region again we have parentheses Embrace and we want to join because this is an or we're going to join these two regions so we're going to have brace negative 2 down to negative infinity and parentheses up to positive Infinity for four now to write the interval notation it's going to look just like the graph with a union symbol in the middle to join the two regions so we're going to have negative Infinity sorry parentheses negative Infinity comma negative 2 brace bracket bracket Union parentheses 4 comma Infinity parentheses now write the interval notation and graph x is less than six X is greater than 0. so that means 0 is less than or equal to X number line bracket for equal sign 6 down to Infinity with a bracket 0 to positive Infinity with a bracket so we've shaded our region again because this is an and we want the overlap so interval notation is just going to be the two brackets and the interval between 0 and 6. now what happens if we give you the interval notation and we ask you to draw the region so there's our number line so how do you draw this well you're going to put your bracket your parentheses and then you're going to shade your regions so we have the region between the three and the nine shaded and the region from negative Infinity to negative three shaded and that's going to be it next how do we solve this inequality first we're going to distribute the 4. so we get 4X minus 4 minus 5 is greater than 9x Plus 1. combine the four and the 5 and we get 4X minus 9 is greater than 9x Plus 1. now most of us remember that if you divide by a negative sign then you flip the inequality sign but I don't like to hedge my bet so I always make sure that I always have positive x's and I never divide by negative signs that way I don't have to remember those Freaky Deaky rules that rarely come up even though I know them I don't like to have to remember them so I'm going to subtract 4 and add one so if I subtract 4 and subtract 1 from both sides then I get negative 10 is greater than 5x then if I divide by 5 I get negative 2 is greater than x writing the interval notation I get parentheses negative Infinity comma negative 2 parentheses now let me know if you have any questions thank you [Music] thank you