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Problem 1. (10 points) Solve the inequality and express the solution in interval notation.X -2 <0 X+2Answer: Solution set $Use 'inf' for 00 , '~in...

Question

Problem 1. (10 points) Solve the inequality and express the solution in interval notation.X -2 <0 X+2Answer: Solution set $Use 'inf' for 00 , '~inf' forand "U' for union.

Problem 1. (10 points) Solve the inequality and express the solution in interval notation. X -2 <0 X+2 Answer: Solution set $ Use 'inf' for 00 , '~inf' for and "U' for union.



Answers

Solve the inequality and express the solution set as an interval or as the union of intervals. $$0 < |x-2| < \frac{1}{2}$$

Okay, so it is easier for me to mhm to show you how the how the solution will be Okay in this little number. A lot. Okay, so What we want to solve is expert is 1/2. It's greater than zero but smaller than two. Okay. Yeah, so the absolute absolute value of X minus what half means? The distance from X. Yeah. To one half. Okay, so this inequality means the distance From X to one half is greater than zero and smaller than two. Okay, so let's find one half. Here is one half right? The distance where half is Greater than zero. Okay, so this one half is excluded, right? This point is excluded. Okay but it is smaller than two. Okay, 12 two. Is is also excluded here. We Count 1 2. Okay so the solutions that will be My last 3/2 comma one half union, 1/2 come a 5/2.

Okay, we're solving this absolute very inequality here. Absolutely. If X is greater than zero but less than one half. Okay, let's deal with this side of the inequality first. That would be everything in between negative one half of one half. If we deal with the other half of the inequality this half, Okay, this would split into X is greater than zero or excess less than I guess negative zero. Okay, this doesn't really make any sense. That's the same as X is less than zero. So basically this is just saying X is greater than zero or less than zero. In other words X cannot equal zero. So if we combine these two kind of options together, all we're doing is taking the zero out of this first number set. So we'll be left with everything from negative one half. 20 Jump over zero using the union and then we'll go out to positive one half. To complete our solution here. Well, trickier there.

Yeah. Okay. Now that absolutely inequality for us to solve here, we're gonna start by using this role at the top, which says we can split this or trapped this in between the positive version of that right hand side and the negative version of that right inside. Okay. From there we add to all three sides. Okay? Which would give us 3/2 is less than X, which is less than 5/2. Or in other words, 3/2, comma 5/2 with Prince sees on both sides as simple as that.

Okay, we're solving this absolute better inequality using the top rule here again. So we're gonna trap this X minus one half. Make it less than two, but greater than negative two. Well, then add one half to all three kind of sides of this inequality give us negative 3/2 is less than X, which is less than 5/2. Or in other words, negative 3/2, 2, 5/2 is our solution first.


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