Integer Inequalities with Absolute Values

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  • 0:01 Absolute Value
  • 0:35 Integer Inequalities
  • 5:32 Example 1
  • 6:11 Example 2
  • 6:35 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will know the ins and outs of solving inequalities involving absolute values. Learn how you will end up with two different answers.

Absolute Value

This video lesson adds to our understanding and knowledge of working with absolute values. Recall that an absolute value is the distance from 0 of a number. The absolute value of any number is always equal to the number without any negative signs. So the absolute value of 9 is 9, and the absolute value of -9 is also 9. The symbol for the absolute value is two pipe lines on either side of our number. So |9| = 9 and |-9| = 9.

Integer Inequalities

In math, you know how we solve equations such as x + 7 = 10 for the unknown variable? We can also solve inequalities that way, such as x + 7 < 10. But remember that, for inequalities, we get a range of numbers for our answer.

For example, solving x + 7 < 10 gives us x < 3. So our answers include all the numbers less than 3. We solved it by subtracting the 7 from both sides to get the x by itself. We used the same process as we would use to solve our standard equations.

Well, now we can introduce inequalities with absolute values. Solving these types of problems uses the same techniques as solving our equations and inequalities, but they do have a few interesting twists to them. Let me show you.

Let's say we have this problem.

|x - 1| < 3

The problem looks easy enough. If we didn't have the absolute value there, we would just add the 1 to both sides to solve our problem. Well, we do have the absolute value there, and we have an inequality. Because we have the absolute value there, we need to create two problems to solve out of our one because we can have two different absolute values equaling the same thing. Remember |9| = 9, but so does |-9| = 9.

Since we have a variable to solve for, we will write two inequalities without the absolute value. So we have (x - 1) < 3 and -(x - 1) < 3. For the second inequality, we can multiply by -1 on both sides to move the negative sign over. Remember that whenever you multiply or divide by a negative in an inequality, your inequality sign will flip.

So now we have x - 1 > -3. Our two inequalities are now x - 1 < 3 and x - 1 > -3. A good way to remember this part of solving an inequality with an absolute value is to just remember to set up two inequalities, one exactly the same as your initial equation but without the absolute value, and the other one with the inequality flipped and the number with a negative sign.

Now we can go ahead and finish solving like we normally do. We add 1 to both sides of our inequality for both inequalities. We get x < 4 and x > -2. We can write our final answer as -2 < x < 4.

Now what if our problem had a greater than symbol instead?

|x - 1| > 3

We will go about solving this problem the same way as if we had the less than symbol. We create two inequalities, one the same just without the absolute value, and the second with the inequality flipped and the number with a negative sign. We have (x - 1) > 3 and (x - 1) < -3. Now we can go ahead and solve each inequality. We add 1 to both sides for both inequalities. We get x > 4 and x < -2. This time, though, we have to write our answer like this: x > 4 OR x < -2.

The first thing you need to keep in mind when solving inequalities with absolute values is that you need to create two inequalities to solve from your one problem. Your first inequality is your problem without the absolute value, and your second inequality is the same inequality but with the inequality flipped and the number with a negative sign.

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