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

|x + 3...

Question

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

|x + 3| ≤ 4

Accepted Answer

Hello today we're going to be simplifying the absolute value inequality and we're going to represent the solution in interval notation. So the inequality given to us is the absolute value of X plus two less than or equal to three. Now, in order to start solving this, we need to go ahead and open up this absolute value in order to do that. This can produce two types of inequalities. The first one where the quantity is X plus two less than or equal to three and the other one where the quantity X plus two is greater than or equal to negative three. So what we need to do now is go ahead and solve both of these inequalities Starting with the left hand side in order to sell for X. We just need to go ahead and some track two from both sides of this inequality that's going to leave us with X less than or equal to 3 -2 which is one. And on the right hand side we're gonna go ahead and do the same thing. So we're gonna minus two from both sides of this inequality. And that's going to leave us with X greater than equal to negative three minus two which is negative five. Okay, now we want to go ahead and represent both of these numbers in set builder notation but to get a good idea of what we're looking at, let's go ahead and draw a number line and on this number line we're gonna go ahead and label the two points that we got which is one and negative five. So negative five is going to be here And one is going to be here. Now what this is saying is we have X greater than or equal to negative five. So X could be any number that comes after negative five, but it also includes negative five because it could be equal to negative five as well. So we're going to go ahead and include negative five in our solution set. And for the other point, this is saying that X is less than equal to 21. So X is all the values that are less than one, but it can also include one because X can be equal to one. So we're gonna go ahead and include one in our solution set. And since we on all the values of X to be less than one and greater than negative five, we want all the values that come between negative five and one. So all of our solutions are gonna be between negative five and one, but we're including negative five and one. So how do we represent that? While whenever we include numbers in a solution set, we go ahead and use our brackets. So since we're including negative five, we're using a bracket for negative five. And since we're including one, we're using a bracket for one. And this is going to be our solutions represented in interval notation. And that also means that the answer to this problem is going to be a so I hope this video helps you understanding how to represent the solutions in set interval notation, and I'll go ahead and see you all in the next video.

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