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Absolute Value Inequalities

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Today we're gonna talk about absolute value inequalities. So we're gonna start by reviewing the idea behind absolute value. And then through examples we'll show how you solve an absolute value expression with an inequality. Absolute value is defined as the distance that a number is away from 0 on a number line. So if I wanted to find the absolute value of 3, that answer would just be 3 because 3 is 3 units away from 0 on a number line. And the absolute value of negative 3 is also 3, because negative 3 is 3 units away from 0 on a number line. So your absolute value is always gonna evaluate to be a non-negative number.

So let's start by looking at less than absolute value inequalities. For example I have the absolute value of x is less than 5, which means our solution set is gonna be all points that are less than 5 units away from 0 on our number line. So that's gonna include any value that's less than 5 units away from 0 in the positive direction, and in our negative direction. We know that it doesn't actually include 5 or negative 5, but we can see that it's going to be everything in between. So x can be any value between negative 5 and 5, and so we can write this as a compound inequality. So negative 5 is less than x which is less than 5. And any less than absolute value inequality can be written as and compound inequality like this. So we could also write this in set notation. Where we have x is all values such that negative 5 is less than x, which is less than positive 5. And we could write it in interval notation. So negative 5 is my lower bound, but not equal to negative 5, and positive 5 is my upper bound.

Let's do another example. I've got the absolute value of 3 x plus 6 is less than 9. So since this is a less than absolute value inequality, I know that the value of 3x plus 6 is gonna be bound between negative 9 and positive 9. So I can write this as an and compound inequality, showing that 3x plus 6 is between negative 9 and 9. And now I can solve this. Gonna start by subtracting 6 in the middle to cancel, as well as on the other side of both of my inequality symbols. So I'll be left with 3x is less than 3, but greater than negative 15. Then I'm going to divide by 3 in the middle, and on each of the ends, cancel in the middle, and I'll have x is less than 1 but greater than negative 5. So I can go ahead and graph this solution on my number line. So again, I want to show that x is in between, but not equal to, either negative 5 or 1. So I have an open circle at negative 5, an open circle at 1, and my solution set is anything in between those. So I could write this in set notation, which would be all x values such that x is greater than negative 5 but less than 1. And I could write it in interval notation, with negative 5 be my lower bound to not equal to negative 5, and similarly 1 is my upper bound but not exactly equal to 1.

Let's talk about greater than absolute value inequalities. So for example, the absolute value of x is greater than 3. So my solution set is gonna be all points that are greater than 3 units away from 0 on my number line. And that includes the values that are greater than 3 units away from 0 in the positive direction, and greater than 3 units away from zero 0 the negative direction. So if I plot that on my number line, I know that again I'm not actually including my exact values for 3 or negative 3, but my solution can be greater than 3 and less than negative 3. So x can be any value that's to the left of negative 3 or to the right of positive 3. And now we can write this as an or compound inequality. So we have that x is less than negative 3, or x is greater than positive 3. So any greater than absolute value inequality can be written as an or compound inequality that looks like this. We could also write this in set and interval notation. So set notation would be, all values of x such that x is less than negative 3 or x is greater than 3. And in interval notation, our first interval is gonna start at negative infinity up to, but not including, negative 3. And our second part of our interval will start at 3, but not including, and go all the way up to positive infinity. And we want to show that our set is actually the union of these two individual intervals.

Alright, for my next example, I've got the absolute value of 5 x minus 15 is greater than or equal to 10. So since this is a greater than absolute value, I know that my value for 5x minus 15 is gonna have to be less than or equal to negative 10, or greater than or equal to positive 10. So I can write that as an or compound inequality, so I've got 5x minus 15 is gonna be less than or equal to negative 10, or 5x minus 15 is going to be greater than or equal to positive 10. So I'm gonna solve each of these inequalities. Going to start by adding 15 to both sides, this will cancel, and I'll be left with 5x is less than or equal to positive 5 , can divide by 5 on both sides. This will cancel, and I'm left with x is less than or equal to 1.

Solving this inequality, I'm gonna start again by adding 15 on both sides. This will cancel, 5x is greater than or equal to 25. I'm gonna divide by 5 on both sides. this will cancel, and I've got x is greater than or equal to 5. So my solution set is x is less than or equal to 1, or x is greater than or equal to 5. So graphing that on my number line. I know I'm gonna have closed circles to include the values of both 1 and 5, so closed circle on 1, closed circle on 5. And then we can have everything that is to the left of 1, and everything that's to the right of 5. So writing that in set notation, I'll have all x values such that x is less than or equal to 1, or x is greater than or equal to 5. And in interval notation, this first interval goes from negative infinity all the way up to and including positive 1. And this interval starts at and including 5 going all the way up to positive infinity, and we'll use a u to show that our solution set is the union of both of those individual intervals.

So let's go over our key points from today. We saw that less than absolute value inequalities can be rewritten as and compound inequalities, where our expression and our absolute value sign is bound between the negative and positive values of this quantity. Greater than absolute value inequalities can be written as or compound inequalities, where your expression inside your absolute value sign is going to be less than the negative of this quantity, or it's going to be greater than the positive value of this quantity. And again, this of course will work for greater than or equal to. So I hope that these examples, and key points, help to understand a little bit more about absolute value inequalities. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.