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 solving the given absolute value inequality. Then we're going to express the answer in interval notation. So what we are given is the absolute value of X plus two. And that is going to be less than or equal to three. Now let's quickly recall one thing about the absolute value graph. Whenever we are looking at an absolute value graph, there's always going to be a positive and a negative portion to the absolute value graph. So when we're trying to solve an absolute value inequality, in order to get rid of the absolute value signs, we're going to need to split up the absolute value inequality into two separate inequalities. For the positive portion of the inequality, we're going to have X plus two is less than or equal to three. And for the negative portion of the inequality, we're going to have X plus two is greater than or equal to negative three. So now that we've split up this inequality into two portions, we need to go ahead and solve each of the portions on the left hand side. In order to solve for X, we're going to subtract two to both sides of this inequality. And this is going to leave us with X is greater than equal to negative five. On the right hand side, we're subtracting two to both sides of the inequality. And that's going to leave us with X is less than or equal to positive one. So now that we have the values X is greater than equal to negative five or X is less than equal to one. Let's go ahead and plot this on a number line. So we have this number line and it's going to contain the two values that we've simplified. We have negative five and we have one and one thing to keep in mind is that we are using the greater than or equal to assemble and the less than or equal to symbol since these values of X can be equal to negative five or one, that means we're going to include these values in our solution set. And we're gonna represent that as a solid dot on the number line. Next, we want all the values that are going to be greater than negative five and we want all the values that are less than negative one. Well, the region that satisfies both of these conditions is going to be the region between negative five and one. So that means our solution is going to include all the values between negative five and one, including negative five and one itself. So now that we have a range of values, let's go ahead and express the solution in interval notation. So since this set, or since we're reading this from left to right, we're going to start at the left most point which is going to be negative five. And since we are including negative five in our solution set, we're going to assign it a bracket, then we want to go ahead and write the end of the solution set, which in this case is going to be positive one. And since we are including positive one in the solution set, we're going to assign a bracket as well. And this is going to be the solution to the given inequality. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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In Exercises 59–94, solve each absolute value inequality. |x + 3| ≤ 4