Absolute Value Inequalities How To Solve It

in this video we're going to talk about how to solve absolute value inequalities so let's get right into it let's say if we have the absolute value of 2x plus 5 and it's greater than 11. so quick and simple how can we solve this what recommend doing is to write two equations or two inequalities the first one is going to look exactly the way you see it without the absolute value symbol the second one here's what you need to do rewrite the stuff inside of the absolute value equation change the direction of the inequality and change the sign of the number that you're dealing with so instead of 11 we're going to have negative 11. now whenever you have greater than symbol this is going to be an or expression if there's less than symbol this will be an and expression and you'll see that throughout the problems that are coming up and now let's go ahead and finish this problem so let's solve the inequality on the left by subtracting both sides by five eleven minus five is six next let's divide both sides by two 6 divided by 2 is 3. so our first answer is is greater than 3. now let's get the second answer so once again we're going to subtract both sides by 5. negative eleven minus five is negative sixteen and then we're going to divide both sides by two negative sixteen divided by two is negative eight so is less than negative eight and if you want to you can check your answers for instance if we pick number that's greater than three will this equation be true or false let's say if we plug in four this would be two times four plus five two times four is eight eight plus five is thirteen and the absolute value of thirteen is thirteen so thirteen is indeed greater than eleven that works so what if we plug in number less than negative eight let's say negative nine will this be true statement two times negative nine is negative eighteen and negative eighteen plus 5 is negative 13. the absolute value of negative 13 is positive 13. and so that is true statement so we can see why these answers are indeed correct but that's how you can check your work if you want to now let's talk about how we can graph this on number line so let's say this is 0 and negative 8. so we have is greater than 3 but not equal to it so we're going to draw line to the right and because it's not equal to 3 we're going to use an open circle now for this one is less than negative 8. because it's less than we're going to draw the arrow to the left all the way to the left we have negative infinity and all the way to the right we have positive infinity so now we can represent our solution using interval notation so could be anywhere between negative infinity to negative 8 as we see here and then union it could be anything between positive 3 and infinity with infinity symbols always use parentheses if you have an open circle you also need to use parentheses if you're dealing with closed circles then you need to use brackets now let's try another problem so let's say we have 3x minus 4 and let's say this is instead of greater than we're going to say it's less than 17 let's say less than or equal to 17. go ahead and try this problem so let's write two equations or two inequalities so the first one is going to look exactly like the original without the absolute value symbol now because it's less than we're going to have the word and instead of or if it was greater than we would have the or situation now for the next one we're going to rewrite what we see inside the absolute value equation or symbol and then we're going to change the inequality and then we're going to change 17 to negative 17. and then all we need to do is solve these two inequalities so let's begin by adding 4 to both sides 17 plus 4 is 21. next let's divide both sides by 3 one divided by three is seven so we get first answer is less than or equal to seven now let's get the second answer so let's add four to both sides negative 17 plus 4 that's going to be negative 13. next let's divide both sides by 3 and so we get is greater than or equal to negative 13 over 3. now let's go ahead and plot what we have so let's draw number line 7 is going to be on the right side negative 13 over 3 that's somewhere to the left negative 12 over 3 is negative 4. negative 13 over 3 that's like negative 4.33 now let's plot this one first so is less than or equal to 7. because it's less than we're going to shade to the left and because it can equal 7 we're going to use closed circle as opposed to an open circle now here is greater than negative 13 over 3 and equal to it or equal to it so we're going to use closed circle as well but because it can be greater than negative 13 over 3 we're going to shade to the right so notice that our answer is between these two numbers so we could say that is less than or equal to 7 and it's greater than or equal to negative 13 over 3. so we can write compound inequality to represent the solution now let's talk about some trap questions that you need to be aware of let's say we have the absolute value of 3x plus 5 and let's say that it's less than negative three what is the solution to this particular problem so want you to think about it feel free to pause the video and try this problem now the absolute value of number can never be negative number so we can only get zero or positive number is zero less than negative three what would you say zero is not less than negative three it is greater than negative three because if you draw number line zero will be to the right of negative three the numbers that are greater are on the right of the number line and the numbers that are less are on the left of number line now what if we pick positive number like four is four less than negative three four is to the right of negative three so it is not less than negative three therefore no matter what value of you plug into this expression this will not be true expression so therefore there's no possible solution therefore you could write no solution as the answer to this problem now let's look at another example so let's say if we have the absolute value of 4x minus 3 is greater than negative four what is the solution for this particular math problem now let's look at the possibilities so this could equal zero is zero greater than negative 4. drawn number line 0 is to the right of negative 4 so this is possible now let's say if we get positive number like 8 is 8 greater than negative 4. 8 is also to the right of negative 4 so yes so any positive number that we select will all be greater than negative 4. so every value of that you can plug in this will always be true for any value of if you plug it into this expression it will always be greater than negative 4. you could try looking for value of that doesn't work you're not going to find it so therefore we could say that can be all real numbers to write that as using interval notation you could say it's negative infinity to infinity and on number line basically you're shading the whole thing could be anything so you could say all real numbers or all solution now let's work on one more example problem this one is going to be longer than the other ones and this problem is going to teach you one key lesson about solving absolute value inequalities and here's the question should we write two expressions now or later notice that we have numbers that are outside of the absolute value expression so in this case until you get rid of those numbers you do not want to write two inequalities you need to get rid of those numbers on the left side until you get the absolute value expression by itself on one side of the inequality then you can write two expressions so the first thing we're going to do is subtract both sides by 5. seventeen minus five is twelve next we're going to divide both sides by three twelve divided by 3 is 4. now notice that do not have any numbers outside of the absolute value symbol in this inequality now it is at this point can separate it into two inequalities the original one and then the second one i'm going to reverse the inequality symbol and change the sign from positive to negative now is this an or situation or is this an ant situation because we have the greater than symbol it's going to be the or situation now let's go ahead and solve each inequality so let's begin by adding 4 to both sides and so we're going to have 7x is greater than 8 and then we can divide both sides by 7. so our first answer is is greater than eight over seven for the next one we're going to add four and so we have seven is less than zero and then we could divide by seven zero divided by seven is zero so the other answer is is less than zero so now let's plot this on number line eight over seven that's little bit more than one so is less than zero we could shade that to the left or it's greater than eight over seven so we're gonna shade that to the right don't forget to put your infinity symbols so the solution is gonna be negative infinity to 0 union 8 over 7 to infinity so that's basically it for this video if you like it feel free to subscribe to this channel thanks again for watching