In Exercises 59–94, solve each absolute value inequality.

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Question

In Exercises 59–94, solve each absolute value inequality.

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Accepted Answer

Hello today we're gonna be simplifying the absolute value and equality and we're going to represent its solutions in interval notation. So the inequality given to us is the absolute value of X less than or equal to five. And we're gonna need to go ahead and open up this absolute value here in order to do that. We need to break this inequality into two separate inequalities. The first one being where X is less than or equal to five and the second one is where X is greater than or equal to negative five. Now both of these inequalities are already simplified so we just need to go ahead and express the solution in interval notation. Now, in order to get a good idea of how to do that, let's go ahead and draw a number line real quick. And on this number line we're going to be labeling both of our given points which are five and negative five, negative five being the left most point and positive five being the right most point. Now our first inequality says that X must be less than or equal to five. So at five we want all values of X that are less than five but we can also include five in our in our interval because X can be equal to five. So we're including five in our interval. And on our second interval X must be greater than or equal to negative five. So all values of X must be greater than negative five but it can also include negative five because X may be equal to negative five. So we're gonna be including negative five in our solution. And since we want all values of X that are less than five and all values of X that are greater than five. All of our solutions are gonna lie between negative five and five. So again, all of our solutions are from negative 5 to 5, but we can include negative five and five. When we are including a point into our notation, we go ahead and use the brackets. So since we're including negative five, we put a bracket next to negative five. And since we're including positive five, we include a bracket next to positive five. And this is going to be our set of solutions in interval notation. That also means that the solution to this problem is going to be d. So I hope this video helps you in understanding how to represent this absolute value inequality in interval notation, and I'll go ahead and see you all in the next video.

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