A compound inequality with no solution

solve for X 5x minus 3 is less than 12 and 4x plus 1 is greater than 25 so let's just solve for X in each of these constraints and just keep in mind that any X has to satisfy both of them because it's an end over here so first we have this 5x minus 3 is less than 12 so if we want to isolate the X we can get rid of this negative 3 here by adding 3 to both sides so let's add 3 to both sides of this inequality the left-hand side we're just left with a 5x the minus 3 and the plus 3 cancel out 5x is less than 12 plus 3 is 15 now we can divide both sides by positive 5 that won't swap the inequality since 5 is positive so we divide both sides by positive 5 and we are we are left with just from this constraint that X is less than 15 over 5 which is 3 so that's that constraint over here but we have the second constraint as well we have this one we have 4x plus 1 is greater than 25 so very similarly we can subtract 1 from both sides to get rid of that one on the left hand side and we get 4x the ones cancel out is greater than 25 minus 1 is 24 divide both sides by positive 4 divide both sides by positive 4 don't have to do anything to the inequality since it's a positive number and we get X is greater than 24 over 4 is 6 and remember there was that and over here we have this and and so X has to be less than 3 and X has to be greater than 6 so already your brain might be realizing that this is a little bit strange this first constraint this first constraint says that X needs to be less than 3 so this is 3 on the number line we're saying X has to be less than 3 so it has to be in this shaded area right over there the second constraint says that X has to be greater than 6 so this is 6 over here it says that X has to be greater than 6 it can't even include six and since we have this and here the only X's that are a solution for this compound inequality are the ones that satisfy both the ones that are in the overlap of their solution set but when you look at it right over here it's clear that there is no overlap there is no X that is both greater than 6 and less than 3 so in this situation we have no solution we have no solution