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

- 4|1 ...

Question

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

- 4|1 - x| < - 16

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to simplify the absolute value inequality, negative absolute value of five x minus 11 greater than equal to negative nine and we want to express the solution set in interval notation. So in this case we're working with an absolute value and recall that an absolute value converts whatever number is inside into a positive because it only cares about the magnitude. So in this case we need to account for if the number inside was positive or if the number inside was negative and I'm going to do that by solving the inequality twice. One for if the number was positive and another for if the number was negative. So if the number was positive I would have negative times positive five X -11 Greater than or equal to -9. And if the number was negative I would have negative times negative five X minus 11 greater than or equal to negative nine. So if I focus on solving the positive side or if the number was positive you can see I have this negative on the outside. So I'm going to divide by negative across the inequality. So that changes the direction of the inequality from greater than two, less than So I'm left with uh less than or equal to and my negative nine becomes positive nine. And on the left hand side I have five x minus 11. So I'm going to continue solving this inequality for X. so I'm gonna add 11. Now I have five X less than or equal to divided by five. I'm left with X less than equal to four. So now I've simplified if the number was positive, so I'm gonna do the same for if the number was negative and you can see I have this negative here again, but then I also have this negative on the inside. So if I multiply the negative with the negative, I'm left with a positive. So my left hand side becomes positive five X -11. Greater than equal to -9. That I continue solving the inequality for X. I'm gonna add 11 just as before, but now I have negative nine plus 11. So I'm left with positive too and five x on the left hand side by divided by five And I have five greater than or equal to 2/5. So now I have these two ranges for X. And I can combine these two inequalities where the first one is saying that X has to be less than or equal to four. But my second one is saying that X has to be greater than or equal to 2/5, so X has to be greater than or equal to 2/5. Notice how I just rearranged my second one, saying that X has to be greater than two fits. So this becomes the range of X values or the solution set for the equation or the inequality and we can write this in interval notation. So I have my lower limit 2/5 and that goes to four. And in this case because we have equal two on both sides of X, I can do brackets for both of these, and that becomes my final answer. So when you're choosing your answer, you can see that answer, choice A and B are very similar, but A has the brackets. And because we have those equal to brackets would be the correct interval notation. So hopefully this video helped and see you next time.

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