Solve Absolute Value Inequalities - Overview
Solve Compound Inequalities - And Statements - Example 1
Solve Compound Inequalities - And Statements - Example 2
Solve Compound Inequalities - And Statements - Example 3
Solve Compound Inequalities - And Statements - Example 4
Syracuse University
Solve Absolute Value Inequalities - Example 4


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So in this example, we're gonna be talking about absolute value inequalities. But you're special cases now. If you think back to solving absolute value equations, we were that sometimes have no solution. Well, one movie have no solution. When the absolute value of some expression was equal to a negative number. Let's just say negative seven, because whenever you take the absolute value of a number, it's always going to be a positive. So any positive that you have could ever equal to negative Seventh? Well, of course not. Because seven is negative. It would never be positive. So in that type of case, we would have no solution. When we're solving equations now, it doesn't mean when you solve inequalities that it's automatically no solution. You actually have to sit and think about it for a second. So let's start with our first example. We have the absolute value of X is less than negative. Three. Well, remember, the absolute value of X will always result in the positive value, no matter what value you plug in for Rex. So no matter what, this will always be some positive number. So we we need to know, is any positive number going to be less than negative. Three. Well, that's never going to be the case, because every positive number will always be bigger than negative three. So in this particular case, because this is a false statement, that means our original inequality has no solution. So when you're solving absolute value inequalities again, just like with equations. If you know this, that you have a negative value once your absolute values isolated, there's going to be a special case when it's negative. In this particular case because it's a lesson sign, it's going to be no solution. Okay, well, let's see what happens when we have a greater than sign. So our second example says the absolute value of X is greater than negative. Three. Remember the absolute value of X, no matter what value you plug in for X, will always result in the positive number. So the absolute value of X will be some positive number. Doesn't matter what it is. It could even be zero. But it's gonna be positive. We want to know. Is it greater than negative? Three. Well, isn't every single positive number going to be greater than negative? Three? Yes, it will, even if X was 00 would still be greater than negative three. So this would result in a true statement, which means that no matter what value you Paul again vrx, you're going to be able t o take the absolute value of it, and it will be greater than any negative number, which means it's all real numbers would be the solution. So in summary, once you've isolated your absolute value symbol, if it's less than of particular negative number like an example one it's going to result in no solution as opposed. Teoh is our example for Number two. When the absolute values by itself and it's greater than a negative number, your result will always be all real numbers, so this will always be true. So again, make sure you absolute values by itself first. And then if there's a negative on the other side, that's when you can sit and try and determine should be no solution or all real numbers

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