in this video i want to show you how to express a solution in interval notation so let's say if you're given an inequality let's say that x is greater than 4. let's plot this solution on a number line and then we're going to write the answer using interval notation so let's say this is 0 and 4 is somewhere to the right of zero and all the ways to the right if you keep on going is positive infinity negative infinity will always be to the left now x is greater than four but not equal to four so therefore it does not include four so we need to use an open circle and because it's greater than 4 we need to shade to the right now to represent this solution used in interval notation it's going to be 4 to infinity but because 4 is not included we need to use parentheses so 4 comma infinity this tells us that x is greater than 4 but not equal to 4. and so that's how you can represent this particular solution using interval notation now let's try another example let's say if x is equal to or greater than two how do you think we should represent this solution using interval notation feel free to pause the video plot the solution on a number line then write it using interval notation so let's put 0 in the middle 2 is to the right of 0. on the left we're going to have negative infinity as usual and positive infinity on the right now this time it's equal to or greater than two so we need to use a closed circle as opposed to an open circle but because it's still greater than two we're going to shade towards the right so to represent this particular solution in interval notation we're going to use brackets instead of parentheses to indicate that 2 is included so anytime you have a closed circle it's always going to be associated with a bracket an open circle will always be associated with parentheses and infinity is always associated with parentheses so it's going to be from 2 to positive infinity and that's the answer now let's look at another example let's say that x is less than three try that one feel free to pause the video so once again let's start with a number line let's put 0 in the middle and 3 is to the right of zero and then let's put our infinity symbols which looks like a sideways eight now x is less than three but does not include three so we're going to use an open circle at three but because it's less than we're going to shade this time to the left as opposed to the right now when you need to write the answer in interval notation you should write it from left to right basically just write the way you see it so the first number is negative infinity that's the number on the left the number on the right where the blue line ends is three so it's from negative infinity to three now as we said before you should always use a parenthesis symbol next to an infinity symbol and we have an open circle so that's going to be associated with a parenthesis so this is the answer it's negative infinity to i mean to positive 3. try this one let's say that x is less than or equal to negative one go ahead and take a minute and work on that one so let's start with a number line so here's zero negative one is to the left of 0. let's put the infinity symbols so this time it includes negative 1 so we're going to use a closed circle and because it's less than negative 1 let's shade to the left so the left side has a negative infinity and the right side has negative one so we're just going to rewrite that here negative infinity to negative one now use parentheses always for an infinity symbol and because we have a closed circle this is going to contain a bracket so it's from negative infinity to negative one including negative one now let's work on a different type of example let's go over compound inequalities so let's say x is greater than two but less than or equal to six write the solution on a number line and describe it using interval notation so let's start with zero two is to the right of zero and six is to the right of two so x is greater than 2 but not equal to it so we need to use an open circle and should we use an open circle on 6 or a closed circle what would you say it's less than or equal to six so we're going to use a closed circle at six because it's less than six we need to shade to the left but because it's greater than two we need to shade to the right therefore we have to shade between two and six now how can we represent the solution in interval notation so the left side has a two the right side has a six so it's going to be two comma six but we have an open circle here and we have a closed circle on the right side so for an open circle use parentheses for a closed circle use a bracket so this is the answer 2 to 6 that's how you can represent it using interval notation now let's try another example let's say that x is greater than or equal to negative three but it's less than four feel free to pause the video try that problem so negative 3 is to the left of 0 and 4 is to the right of it so we're going to have a closed circle at negative three but an open circle at four so it's greater than negative three but less than four so we're going to shade to the right of negative three but to the left of four so it's between these two in interval notation the answer is gonna be negative three comma four we're gonna start from the left and end on the right now we have a closed circle here so this is gonna be a bracket and this is an open circle so we need parentheses so that's the answer negative three to four now here's another example for you let's say that x is less than negative two or x is greater than or equal to five try that one so let's start with a number line so let's put zero negative 2 is on the left 5 is on the right and then let's put our infinity symbols now x is less than negative 2. so we're going to have an open circle but because it's less than we're going to shade to the left or x can be equal to or greater than 5. so therefore we have a closed circle but because it's greater than we're going to shade to the right now how can we represent this particular solution using interval notation so let's start from left to right so let's focus on the left side on the left of that portion we have negative infinity and then we have negative two now with infinity you should always use a parenthesis symbol and because we have an open circle there's going to be a parenthesis associated with negative two now to jump from this section all the way to that section and to skip everything in the middle we need to use a union symbol to connect these two together now it includes five we have a closed circle at five so therefore we need to use brackets for five and then for infinity we're going to use an open circle so you can literally see the numbers that you need to use in interval notation but if you have two separated parts connect them with a union symbol and so this is the answer it's negative infinity to negative 2 union 5 to infinity now let's try one more example let's say that x is less than or equal to one or x is greater than two so based on the last example you could try this one so we have zero one and two and positive infinity and negative infinity so first x is equal to or less than one so we need to shade to the left but x is greater than 2 so we need an open circle but we're going to shade to the right now we're going to have the stuff negative infinity 1 2 and infinity so let's start from the left side so negative infinity with parentheses to 1 and this is a closed circle so we're going to use brackets and then union 2 it's an open circle so we need to use parentheses to positive infinity using parentheses and that's it so negative infinity to 1 union 2 to infinity so now you know how to represent a solution in interval notation and the best thing to do is to graph the solution on a number line because at that point you can clearly see what the answer should be in interval notation so thanks again for watching you