in this video we're going to focus on solving compound inequalities we're going to talk about how to graph it on a number line and also how to write the solution using interal notation so let's go ahead and begin here's the first problem let's say that 3x + 5 is less than 8 or -2x + 5 is less than or equal to 1 what should we do in order to find or solve for x we need to solve both equations let's start with the one on the left side let's begin by subtracting both sides by five 5 - 5 is zero we could bring down to 3x and 8 - 5 is 3 next let's divide both sides by three so we can see that X is less than one now for the other side let's begin by subtracting both sides by five so what we now have is -2X is less than or equal to -1 - 5 which is -6 now just like we did before we need to divide both sides by the number in front of X in this case is -2 now you need to keep in mind that whenever you multiply or divide by a negative number you need to change the direction of the inequality -6 / -2 is 3 and so this is what we have now let's plot the solution on a number line so let's say this is zero this is going to be1 here's one 2 and three now X is less than one but not equal to it so we're going to have an open circle and because it's less than we're going to shade towards the left now X is also greater than or equal to 3 because we have the underline because it can equal three we're going to have a closed Circle at three and because it's greater than we're going to shave towards the right so that's how you can graph the solution using a number line now how can we write the answer using interval notation all the way to the left we have negative infinity and all the way to the right we have positive Infinity so the answer the first part is going to start from negative Infinity to one now whenever you have an open circle make sure you use parentheses as opposed the brackets now at three three is included so that's a closed Circle we need to use brackets so that's the answer used in interval notation X could be anything between negative Infinity to one but not including one and anything between three and infinity including three here's another problem that we could try let's say that 7 is less than or equal to 4x + 3 which is less than 23 so here we have another compound inequality the other one was associated with the word or and this one's associated with the word and go ahead and solve it feel free to pause the video so what's the first thing that we should do in this problem the first thing that we can do is subtract all three sides by three if we do that this is what we're going to get 4X is greater than 7 - 3 which is 4 and it's than 23 - 3 which is 20 next we can divide all three sides by four our goal is to get X by itself in the middle so 4 ID 4 is 1 and 20 / 4 is 5 so X is between 1 and 5 now let's plot this on a number line now at one should we have an open circle or should we put a closed Circle notice that X is equal to or greater than one so includes one therefore we need a close Circle at one and because it's greater than one we need to shade towards the right now looking at the other side x is less than five but not equal to it so we're going to need an open circle at five and because it's less than five we need to shade towards the left and we can clearly see that X is between 1 and 5 now how can we represent the answer using interval notation starting at one we need to use a bracket and then after that it stops at five but because we have an open circle at five we need to use parentheses so that's the answer used in interval notation it's simply from 1 to 5 so now here's another problem that you could try 13x + 2 is less than or equal to 4 or 6 - 2 over 5x is less than 2 so take a minute pause the video and try this example so let's start with the one on the left let's begin by subtracting both sides by two so 1/3 x is less than or equal to 4 minus 2 which is two now what should we do at this point to get rid of the X I mean not the X but to get rid of the fraction if you want to get rid of the fraction multiply both sides by three 3 * 1/3 which is 3 / 3 that's equal to 1 or 1 x which is simply x 2 * 3 is 6 so X is less than or equal to 6 now for the other one let's begin by subtracting both sides by six so now we have -2 over 5x is less than 2 - 6 which is4 so now let's multiply both sides by neg five and keep in mind anytime you multiply or divide by a negative number the inequality will change direction so these will cancel and we're going to have positive 2x on the left side and that's greater than4 * 5 which is POS 20 and now let's divide both sides by two x is greater than 10 so let's say this is six and this is 10 and we have the numbers in between so X is equal to or less than six so we're gonna have a closed Circle at six and because it's less than we're going to shade towards the left X is greater than 10 but not equal to it so we're going to have an open circle shaded towards the right at the right we have positive infinity and at the left negative Infinity so the answer using an interval notation is negative Infinity to six since it includes six we're going to use the brackets because we have a closed Circle and then Union 10 to Infinity anytime you have an open circle make sure you use parenthesis and for infinity symbols you should always use parenthesis here's another one that we could try 8 is less than 2 3x + 6 Which is less than or equal to 12 try that one find a solution pled on number line and then write the solution using interval notation so let's begin by subtracting all three sides by 6 8 - 6 is 2 12 - 6 is 6 now we have a fraction so let's multiply both sides by the denominator of the fraction so we're going to multiply all three sides by three 2 * 3 is 6 3 * 2/3 the threes will cancel leaving behind 2x 6 * 3 is 18 next let's divide all three sides by two 6 / 2 is 3 2x / 2 is just x 18 divid 2 is 9 so X is between 3 and 9 so let's say this is three and this is nine now X is greater than three so it doesn't equal three so we have an open circle shaded towards the right X is less than or equal to 9 so it's a closed Circle at 9 shaded towards the left so it's between three and n and in interval notation the answer is going to be 3 to 9 with a bracket at nine since we have a closed Circle at nine so that's it for this video thanks for watching by the way if you want to find more videos you can check out my website video- tr.net or you can go to my channel and you can find a playlists on algebra trig pre-cal calculus chemistry and physics so go ahead take a look at those and uh thanks again for watching this video