Solve each inequality. Give the solution set in interval notation...

Question

Solve each inequality. Give the solution set in interval notation. | 5/3 - (1/2) x | > 2/9

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to express the solution set for the absolute value inequality, absolute value of 3/5 minus one half X greater than nine divided by 10 or 9/10. So in this case we're working with an absolute value. So we want to consider what the absolute value means. So when we have an absolute value, the number inside the absolute value could be positive. So we can have positive 3/5 minus one half X. Greater than 9/10. But it could also be negative because recall the absolute value makes any negative number positive. So we could have negative 3/5 minus one half X greater than 9/10. So in order to find the solution set for this inequality, we want to consider both scenarios. And recall for the solution set, we're trying to find the values of X. That make this inequality true. So I wanted to solve for X. So if I start with my positive side, I'm going to start by subtracting 3/5 from both sides. So it leaves me with negative one half X greater than 9/10 minus 3/5. and if I want to change 3/5 to have the same denominator as 9/10. I need to multiply by 2/2. So that gives me 6/10. So I can write this as 9/10 minus 6/10. So this becomes negative 1/2 x Greater than 3/10. And in order to get rid of the negative one half, I can multiply by -2. So this becomes positive and the two in the denominator cancels out. But I also need to do it to the other side of the inequality. And because I'm passing a negative across the inequality, it switches direction. So now instead of being greater than it will be less than and I have X is less than three times negative, two is negative six Divided by 10. I can write that as negative 3/5. So that is if the number inside the absolute value was positive, if we look at the other scenario where the number is negative, you can see I have this negative on the outside. So I'm going to deal with that first By dividing both sides by -1. And again I have a negative across the inequality. So we will switch directions from greater than two, less than I'll have 3/5 minus one half X less than negative 9/10. And I'm going to subtract 3/5 again. So I have negative 1/2 x Less than -9/10 -3/5. and remember that I said 3/5 was the same as 6/10. So I can write that in here. Now I have negative one half X less than negative nine minus six is negative 15 10th. And to get rid of the negative one half all multiplied by negative two again to both sides. And once again I'm passing a negative across an inequality. So it switches directions. Once again, So it goes from less than two greater than So I have X is greater than negative 15 times negative two is positive Divided by 10 or X Is greater than three. So now I solved both scenarios and you can see I have two inequalities but I need to change them into an interval so I'm going to draw a number line to see how I can combine these two inequalities and I'll say that this is zero, this is negative 3/5. And also to this here is three. So my first inequality is saying that X is less than negative 3/5. So I'll draw parentheses and a line going to the left and a parentheses because there's no equal to and going to the left because I am less than -3/5. And if I look at my second inequality, it's saying that X is greater than three. So I'll do another parentheses at three and now I need to go to the right because X is greater than three. So you can see that there's this separation here, which means I'm going to need a union when I write the interval. So if I look at this first interval, X is going from negative infinity to negative 3/5 And then again there's that union, it's going from three to positive infinity. So this becomes my final interval for the solution set. And if we look at the answer choices, you can see that answer choice A Also has negative infinity to negative 3/5 union with to positive infinity. So hopefully this video helped and see you next time.

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Solve each inequality. Give the solution set in interval notation. | 5/3 - (1/2) x | > 2/9

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