Double inequalities

we're asked to solve for X and we have this compound inequality here negative 16 is less than or equal to 3x plus 5 which is less than or equal to 20 and really there's two ways to approach it which are really the same way and I'll do both of them and I'll actually do both of them simultaneously so one is to just solve this compound inequality all at once and I'll just rewrite it negative 16 is less than or equal to 3x plus 5 which is less than or equal to 20 and the other way is to think of it as two separate inequalities but both of them need to be true so you could also view it as negative 16 has to be less than or equal to 3x plus 5 and and 3x + 5 3x plus five needs to be less than or equal to 20 this statement and this statement are equivalent this one might seem a little bit more familiar because we can independently solve each of these inequalities and just remember the and this one might seem a little less traditional because now we have three sides to the statement we have three parts of this compound inequality but we can see is it we're actually going to solve it the exact same way in any situation we really just want to isolate the X on one side of the inequality or in this case one part of the compound inequality well the best way to isolate this X right here is to first get rid of this positive five that's sitting in the middle so let's subtract five from every part of this compound inequality so I'm going to subtract five there subtract five there and subtract five over there and so we get 16 negative 16 minus five is negative 21 is less than or equal to 3x plus five minus five is three X which is less than or equal to 20 minus 5 which is 15 and we could essentially do the same thing here if we want to isolate the 3x we can subtract five from both sides subtract five from both sides we get negative 21 and negative 21 is less than or equal to 3x + and we get subtracting 5 from both sides and notice we're just subtracting 5 from every part of this compound inequality we get 3x is less than or equal to 15 so this statement in this statement once again are the exact same thing now going back here if we want to isolate the X we can divide by three we have to do two every part of the inequality and since three is positive we don't have to change the sign so let's divide every part of this compound inequality by three you divide every part by three this is equivalent to dividing every part of each of these inequalities every part of these inequalities by three and then we get negative twenty-one divided by three is negative seven is less than or equal to X which is less than or equal to 15 divided by 3 is 5 you do it here you get negative seven is less than or equal to X and and X is less than or equal to 15 over 3 which is 5 this statement and this statement are completely equal and we've solved for X we've given you the solution set and if we want to graph it on a number line if we want to graph it on a number line it would look like this it would look like this is 0 this is 5 this is negative 7 our solution set it includes everything between negative at 7 and 5 including negative 7 and 5 so we'll have solid it we have to fill in the circles on negative 7 and positive 5 and it is everything in between that's our solution set and so we can verify that these works you could try out a number that's well inside of our solution set like 0 3 times 0 is 0 so you're just left with 5 is greater than or equal to negative 16 which is true and 5 is less than or equal to 20 or negative 16 is less than or equal to 5 which is less than or equal to 20 so that works that makes sense you could try 5 if you put 5 here you get 3 times 5 plus 5 well that's just 20 negative 16 is less than or equal to 20 which is less than or equal to 20 that works negative 7 should also work 3 times negative 7 is negative 21 plus 5 is negative 16 so you get negative 16 which is less than or equal to negative 16 which is less than or equal to 20 and you could try other values you could go outside of our solution set try something like 10 10 should not work and if you see here if you put 10 here you get 3 times 10 plus 5 is 35 35 negative 16 is less than or equal to 35 but 35 is not less than or equal to 20 and that's why 10 is not part of our lucien set