Solving Absolute Value Inequalities

In mathematics, an absolute value or modulus is the value of a number without regard to its sign. The absolute value of a real number is also called its magnitude. For example, the absolute value of five is five, but neither –5 nor 5. The absolute value of a number may be thought of as its distance from zero, as a measure of how far it is from zero on a number line. In mathematics, the absolute value (or modulus) of a real number "x" is denoted by |x|. The absolute value of a nonzero real number is greater than or equal to zero. It is also equal to the distance between the number and zero measured in the direction opposite to the negative one. The absolute value of a real number is also called its value, magnitude, or size.

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Video Transcript

at absolute value inequalities. Now these air very much like absolute value Equations accept their inequalities. So let's just start with one. Let's start with the absolute value. Inequality in absolute value of L minus two is less than eight now, just like it was. If it was an equation, we wanna make sure it's simplified and our absolute value is by itself. And then we want to work it twice. Now, there's a little bit of difference when we work it twice, but we'll get to that. So the first time we work it, we're gonna leave everything and all the numbers alone, including the inequality Son, we're just gonna remove our absolute value bars. So l minus two is less than eight, So we'll add to to both sods. So, um, is less than 10 now, The second time I work it, we're gonna go ahead and we're gonna leave the e l minus two alone. We're on Lee going to change what's on the outside of an inequality of the absolute value bars. So that means the inequality sign must change as well. So instead of less than we're gonna put greater than and instead of positive eight, it will be negative. Eight. So both the inequality and the number will change. So we're gonna add to so, um, is greater than negative six. Now, before I can just say that's my final answer. We need to decide if this is an and statement or an or statement. So we have, um he is less than 10, and we have l'm is greater than negative six. So the only way of really knowing that are the best way of knowing it is to go ahead and let's draw a number one. So I'm gonna do my number line, and we're gonna start negative six. And we're just gonna count by twos. Okay, so let's start with them. Is less than 10. So it's gonna be an open circle, and my arrow is going to be going to the left, and I'll finish that in a minute. Then we have em is greater than negative six again an open circle. And because it's greater than it's going to go to the rot, this means that both arrows are going to meet and that both inequalities air going to share solutions, possible solutions. So this means this is an and statement now, because this is an and statement, we're going to rewrite our answer. Now you can write it with and but both and solutions share the variable l'm. So we're going to rewrite this with that variable. So we're going to start with our lowest numbers, which is negative six and negative six. The arrows pointing towards the narrow native six and negative six is less than M. And then we see that l'm is less than 10. So that is my solution. Let's look at another one. Let's look at the absolute value of X minus two minus five is less than negative too. Now, for this one, we need to go ahead and do some simplifying. So we need to go ahead and we need to add five to both shots. That's going to move it to the right and leave me with just the absolute value of X minus two is less than three. This is what I will work twice. So I will work out. X minus two is less than three. I'm gonna add to to both sides. So X is less than five. And then the next time I work it, I'm gonna have X minus two. And then I'm gonna change once on the outside of the absolute value signs. So I'm gonna have greater than negative three. So we're gonna add to again and this time, ex is going to be greater than negative one. So now let's decide. Is this an and statement or an or statement? So we have X is less than five, and we have X is greater than negative one. So we need to go ahead. We need to draw our number line so we'll just do our odd numbers will count by odd. So let's start X is less than five open circle and it's going to the left. Then we have X is greater than negative one. Go into the right. So this because both arrows air going towards each other, this isn't and statement. So let's rewrite our solution. We're gonna start with our smallest quantity, which is negative. One negative one is less than X, which is less than buff. Okay, now, both of those examples ended up being and statements, but occasionally you actually could come up with an or statements. Let's look at that one. That might be an or statement. Let's look at to times absolute value of 10 B plus seven minus one is greater than 73. So let's go ahead and simplify. So let's add one to both sides. So two times the absolute value of 10 B plus seven is greater than 74. Let's divide by two. So now we have the absolute value of TNB. Plus seven is greater than 37 so this is what will work out twice. So the absolute value of TNB plus seven is greater than 37. So the first time we work it will have 10. B plus seven is greater than 37. We're going to subtract seven on both sides, so 10 B is greater than 30. The Val Bettin B is greater than three. The second time I worked this I'll have 10. B plus seven is less than negative. 37 because remember, those were the parts that have to change. We'll subtract seven. So 10 B is less than negative. 44. We're going to divide by 10, and that will not become a whole number. So let's simplify that fraction. So negative. 40 for over 10. It's still gonna be less than it's gonna be negative. 22/5. So those are my two solutions. So we have B is greater than three, and B is less than negative. 22 over five. So we'll draw our now we're line out, so let's just go. Negative five. Negative four. Okay, so we have b is greater than three. Open circle. So it's going to the right than be Is less than negative. 22/5. That is approximately, um, about four 0.5. So that's gonna be between the four and the five. So I'm gonna draw a circle there, and it's gonna be going to the left. So here we see are two statements. Are arrows air going away from each other? So this is an or statement because the arrows go away. It is an or statement