In Exercises 59–94, solve each absolute value inequality.|3x - 8|...

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 absolute value inequality, absolute value of two X -9 greater than eight. And we want to use interval notation to express the solution set. So in this case we're going to be working with an absolute value. So let's recall what properties and absolute value has. So if I take the absolute value of a number such as a A I get that same number a back. But if I take the absolute value of negative numbers such as negative A. I also get positive a back. So you can see that when I take the absolute value of a negative number the negative number loses its negative sign which means that the number inside an absolute value could be positive or negative. So we want to keep this in mind when we're trying to find our solution set. So if I look at this absolute value, I have two X -9 greater than eight. So in this case two X -9 could be positive or negative. So if it's positive I would have positive two X -9 greater than eight. And if it's negative I would have Negative Times two x -9 greater than eight. So if I solve this inequality for two x minus nine being positive I have two x minus nine greater than eight. I'll add nine to both sides, leaving me with two X greater than and finally divide by two leaving me with X Greater than 17/2. So that's solving the first inequality. And if I go to if two X -9 was negative, you can see I have this negative on the outside which you can say is a negative one. So in order to simplify this I'm going to divide by negative one Across this inequality. So if I do that this cancels out and I'm left with two X -9. And because I divided by a negative across the inequality of inequality changes directions. So instead of being greater than it's now less than and this eight is now negative eight because eight divided by negative one is negative eight but I can continue doing my simplification so I can see this is by ad nine, I now have 2x less than positive one. And finally, if I divide by two I have X is less than one half. So these are my two intervals and if I combine them I have my upper limit is 17/2 because 17/2 is greater than one half. But it seems like there's no upper limit because X is greater than 17 House. So if I draw a number line and say this bottom line is 1/2 and this upper line is 17 house For this first interval where X is greater than 17 house, I have a parentheses because there's no equal two and X goes off to the right, so to positive infinity. And for the second inequality I have X less than one half. And once again it's the parentheses because I don't have an equal to and now X is less than which means X goes off to the left, so X heads off to the left or too negative infinity. So now I have this sort of range and you can see that there's no overlap. So I need to do a union for both of these intervals. So I have X going from negative infinity Up to positive one half, Union with 17 half House to positive infinity. So this represents my solutions that in interval notation, and you can see that this aligns with answer choice A. So hopefully this video helped and see you next time.

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In Exercises 59–94, solve each absolute value inequality.|3x - 8| > 7

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