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Absolute Value Inequalities
Absolute Value Inequalities. This is a relatively rare topic on the test. So it doesn’t often appear, but it may appear. Especially on one of the harder problems. If you haven’t seen the introduction material on absolute value in the lesson Positive and Negative Numbers 2 in the Arithmetic and Fraction module, I strongly suggest watching that lesson before you watch this one.
So from that lesson recall the distance definition of absolute value. Absolute value is the distance of x from 0. And of course distance is always positive, which is why the absolute value is always positive. The absolute value x minus 5 is the distance of x from 5. Absolute value of x plus 3 is the distance of x from negative 3.
This way of thinking about absolute value is enormously helpful in understanding inequalities that involve absolute values. Here’s a practice question. Pause the video and then we’ll talk about this. Okay.
Express the absolute value of x minus 7, is less than nega, is less than or equal to negative 3 as an ordinary inequality, inequality not involving the absolute value. Well, we start off from 7 on the number line, and we can go a distance of 3 from that starting point. We can go a distance of 3 in either direction.
So when we go down 3, we get to 4. When we go up three, we get to 10. Those are the two end points. Because it’s less than or equal to, that means the end points are included. So, x can be 4 and x can be 10. Those are the end points, and so everything between them is allowed.
And so this would just be 4 is less than or equal to x is less than or equal to 10. So that is, that re-expresses that absolute value inequality as an ordinary inequality. Here’s another practice question. Pause the video and then we’ll talk about this. Express the region on the number line as an absolute value inequality.
So as an ordinary inequality that would be easy. As an ordinary inequality this is just x is greater than 20 and less than 90. But that’s not what we’re being asked. We’re not asked for an ordinary inequality, we want an absolute value inequality. The first thing we need to do is find the center of that region.
The middle of the region is the average of the end points, 20 and 90, so that middle is 55. So 55 is the center. Now the question is, how far do we move away from that center? Well, 90 is 55 plus 35, 20 is 55 minus 35. So we start at that center, 55, and we, from 55 we can go a distance of 35 in both directions.
Notice also that the end points are not included, so we can go, we cannot go as far as 35. We have to go a distance that is less than 35. So in other words, the distance from 55 is less than 35. And that is the absolute value inequality that exactly expresses that particular region.
Here’s another practice question. Pause the video and then we’ll talk about this. Okay once again the first thing we have to do is find the middle of that region, we find the middle of the region by averaging. So negative 3, plus positive 11 is positive 8 divided by 2 is 4.
That’s the middle point. So, if the midpoint is 4, how far do we go in either direction? Notice that x can go as far as 7 above 4, which is 11, or 7 below 4, which is negative 3. Notice also that those endpoints are included, so the distance can be less than 7.
The distance can also be equal to 7. It can go a distance of 7. Away from 4, and that’s still included. So the distance from 4 is less than or equal to 7. And that’s the absolute inval, absolute inequality expression of that same region. We demonstrated how the distance understanding of the absolute value makes inequalities involving absolute values very easy to solve.
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