Unit 8, Lesson 16
Solving Inequalities with Absolute Values
Integrated Math 1
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The part that winds around on a bolt is called the “thread.” In order to work correctly, the threads of a certain type of bolt must be about 1 millimeter apart from one another. Due to the way they are made, it is difficult to get the threads exactly 1 millimeter apart, but they will not work correctly if they are too far off. This type of bolt is acceptable if the threads are within 0.1 millimeter of what it is supposed to be.
Use the number line to draw all of the distances between threads that are allowed.
Let represent the distance between threads. Complete the two inequalities that describe what must be true for all bolts of this type that are acceptable.
and
Graph the solution to each inequality on a number line.
In many real-world situations, it is okay if an item is not exactly the perfect length or weight. There is a range of values that will work: The length or weight of the item can be within a certain amount or at least a certain amount away. In these cases, it can be useful to write the range of values that work as inequalities that include absolute values.
To be within a certain distance of a value, we can write an inequality of the form . The solutions to this inequality can be drawn on a number line like this:
To be at least a certain distance away from a value, we can write an inequality of the form . The solutions to this inequality can be drawn on a number line like this:
For example, to fence in a circular area with a radius of 100 meters, we should use the equation and buy meters of fencing. This is difficult to do because is an irrational number. Maybe it’s okay if we don’t make a perfect circle or if it’s a little off, so if we buy amount of fence, where is a solution to , we will have the right amount of fencing within 1 meter of the exact value, and that should be close enough.