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

|3x - ...

Question

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

|3x - 8| > 7

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to simplify the inequality, absolute value of two X minus nine, creator than eight. And we want to express the solution set for that inequality and interval notation. So in this case we're working with an absolute value and recall that an absolute value makes the number inside positive. So the number inside the absolute value could really be positive but it could also be negative. So when we're solving for this inequality I'm going to take into account both of those scenarios. So the number was positive, I would have positive two X minus nine greater than eight. But if the number was negative I would have negative times two X minus nine greater than eight. So I'm gonna work on solving if the number was positive first. So I'm going to add nine to both sides. So I'm left with two X greater than 17, Divide by two. I'm left with X greater than 17 divided by two. So that simplifies if the number was positive. Now I'm going to look at if the number was negative, going to divide this negative and notice that I'm dividing by a negative across inequality. So my inequality switches directions. So now I have two X -9, Less than -8. If I continue simplifying and when I add nine to both sides, so I have two X, less than one, divide by two. I have X is less than one half. So now I have these two inequalities that represent the range of X values that will satisfy the inequality absolute value of two, X -9 is greater than eight. So in this first inequality, x greater than 17 house you can see that the lower boundary is 17.5 because it's saying that X has to be greater than that and there's no upper boundary. So I'm just going to say positive infinity because We just need to make sure that X is greater than 17/2. So now I'm going to look at my second condition which is saying X has to be less than one half, which is saying that 1/2 is the upper boundary and X just needs to be lower than that. So negative infinity is the lower bound. So you can see that these are my two intervals for the values of X that will satisfy the inequality. So I need to do a union between these two. And if I do that I'm left with negative infinity to one half, union with 17 Halfs to positive infinity. And this becomes my final answer for the solution set that will satisfy two X -9 greater than eight with 2 X -9 being in an absolute value. And you can see that this aligns with answer choice A So hopefully this video hoped and see you next time

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