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5487: Proposed by Kenneth Korbin, New York, NX+1)4Given that T(:with <Find positive integers andVh -b +...

Question

5487: Proposed by Kenneth Korbin, New York, NX+1)4Given that T(:with <Find positive integers andVh -b +

5487: Proposed by Kenneth Korbin, New York, NX +1)4 Given that T(: with < Find positive integers and Vh - b +



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If $q$ is a negative number, $r$ is a negative number, and t is a positive number, determine whether each expression simplifies to a positive or negative number. If it is not possible to determine, state so. $$ q+t $$

So with this problem, we have, um, thes different positive integers and we have tee boxes. Eso we'll have, we'll say. And this is the teeth box. This is the first box, second box, third box, and then we have the teeth box, and then we have the number of objects. So if the number of objects we have, the smallest number we could possibly have is on one B one and two b two and three b three and so on. So with that, um, since the final object is nt minus, T plus one will say that n t N t could be equivalent. And then plus, one would mean that there is that one extra object. Um, and ultimately, what that's going to mean is that we have all these objects and this could be yeah, and one and two and three. And then we get to the very last one NT minus t plus one. So we have an object that goes in there, um, and then, But we'll end up having is we'll have to keep going through it again and again and again. But the and the the ice box. So we'll say this is the icebox or something like that. We know that it will contain and I objects because if this was and three, then that's the end. Three objects. If this if this was the icebox, then this would be and to objects. But it would also because here would be in one and two and so on, and that's the smallest case. Again, we could have even larger numbers in which there would definitely be at least that. But because it's at least we know that this is going to be the case.

Let's figure out the sign of tea plus R t is positive. Are is negative. What do you get for the sign of a positive plus a negative? Well, it depends. Here's a couple of examples. Suppose you're positive. Number was three and your negative number was negative, too. When you add those together, you get a positive. But suppose you're positive. Number was two and you're negative. Never was negative. Three. When you add those together, you get a negative. So it depends on which numbers you're adding. So just by talking about positive and negative, it is not possible to tell what you get when you add them, you have to know what numbers they are.

Let's figure out the sign of tea times the quantity of Q plus our where t is a positive and cue. Plus, our is a negative plus a negative. Okay, Working inside the parentheses, a negative plus a negative will be negative. And we're multiplying that by the positive on the outside and a positive times, a negative will be positive. So our answer is this expression is positive.

In an absolute value inequality problem. If you have a greater than symbol at the beginning, then you are going to have an or statement. If you have a less than it isn't and so on, the 1st 1 you will start out with T is greater than or equal to zero. And then we will put yeah, t is less than or equal to zero. When is that true? Well, great of Enrico. Zero means zero positive, less than or equal to zero means zero or negative. Well, that's everything. That's all negatives, all positives and zero. So that's all real numbers for negative infinity. The positive infinity. The second part starts out as a less than one. So this is gonna be an end statement. T is less than or equal to zero and T is greater than or equal to zero. The first statement is basically saying that we have a number that is zero or negative. Second condition is that tea is zero or positive. Well, there's only one way to Fort to fit both, and that's if t is zero. So we will simply say t equals zero


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