Question Video: Estimation of Real Numbers Involving Roots to Compare between Two Numbers

Is −1 + √2 greater than, equal to, or less than −√(2 + 1)?

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Video Transcript

Is negative one plus square root two greater than, equal to, or less than negative square root two plus one?

So notice both of these expressions have a one and a square root two. They just switch the signs from each term. So in the first one, here the one is negative and the square root two is positive, where in the second one the one is positive and the square root two is negative.

So we wanna know is the first one, the first expression, greater than, equal, to or less than this second expression. So we have to decide. One and square root two, how are they related in terms of size?

Here we’ve put a couple square roots on a number line. Here’s our square root of two. Now we just need to figure out where is one. Well the square root of one is one. So it’s here. It’s less than the square root of two. So one is smaller than the square of two.

So in our first term, the one is negative. So the smaller number’s negative. And the larger number’s positive. So since the larger number is positive, these two added together is positive.

Now looking at our second expression, the larger number’s negative and the smaller number’s positive. So we would get a negative number. So a positive number is greater than a negative number. Therefore, our answer would be greater than.

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