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Ptcve #hue f~Lzwicz inequality:For V "[ejl * sigk</-8...

Question

Ptcve #hue f~Lzwicz inequality:For V "[ejl * sigk</-8

Ptcve #hue f~Lzwicz inequality: For V " [ejl * sigk</-8



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Prove the triangle inequality $\|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\|$

Being equality one divided by three V plus 1/4 B is less than 1/2. Well, the first thing to do is find out what this restricted in denominator. You see that we cannot equal zero because three recon on he was 04 week equals zero. So now what we can to is turn this inequality into the equation and software. We got one over TV. It was one of the four. V is you go to 1/2 if you multiply both sides of the equation by 12 V, which is the least common denominator Bob. Okay, there's 12 v times 1/3 V goes 12 v times who won over 40 plus one is equal to 1/2 times 12 v. So our equation is foreclose. Three is you 26 TV, where he is equal to seven, divided by six. No, if we look at our number line here and we exclude the values of zero and seven or six her well, don't wait around. My apologies. We have zero here and we have 7/6 here. We can plug in different numbers from each interval and see if the original inequality works. So if you play gin, that's a negative one. No, no. One. And to hear let's will be kept repugnant. A negative one here because negative 1/3 plus negative. 1/4 assessing 1/2 and a factor Negative. 7/12 iss less than what half. So we accept this multiple, I'll be plug in vehicles one, We got 1/3 plus one for this lesson. One happened, but 7/12 is not less than 1/2. So we saying no to this interval here. And if we plug in vehicles, too, you got 16 plus 1/8 is the same 1/2 In fact, 7 24 is less than 1/2. So we accept this interval and our answers are vey is less than zero when distant trouble here. And he is greater than 76 from this interval here.

Let's recall the cushy Schwarz inequality, which says that the absolute value off dinner product you against V is lower equal, then the product off the norms. Now, part of your case, we know what you and we are. So let's start by computing you against feed and these Amir, it's just a plus B. Now let's compute the length of you. Let's compete squared so that we don't have to worry about the route. This is just a square plus B squared, and then the norm of V squared is too. Now, since we have the the normal view and V squared, we can take the Christian Schwarz inequality and square everything because all the terms are beating our positive numbers. So let's put the square is everywhere the inequality still valid, so we can write these as a plus B. All square is lower equal than to a squared plus B squared. Now we're almost there. Let's divide everything by four. So on the left inside, while the four gets inside the brackets, so it becomes able to be half everything squared. And then we have a lover equal a squared plus B squared, divided by two as we wanted to see

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

Okay, we're going to solve this compound inequality. And first, I think it would be easier to work with fractions instead of decimals. That might surprise some people, but it helps you keep things exact. So I'm going to change my 0.75 and call it 3/4 and my 4.5 and call it nine hats. Now, in order to isolate V, we need to get rid of the multiplied by 3/4 1 idea would be to divide by 3/4 but another idea would be to multiply by 4/3. So if I multiply that by 4/3 I need to multiply the left and the right by 4/3 as well. So now we have 4/3 times, too, which is 8/3 is less than or equal to be. Because the 4/3 cancels with 3/4 is less than or equal to now. This next one actually reduces because the nine over three reduces we get a three and a one on the four over to reduce is we get a two and a one, so that would be just less than or equal to six


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