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Problem 41 Easy Difficulty

How close to -3 do we have to take $ x $ so that
$$ \frac{1}{(x + 3)^4} > 10,000 $$

Answer

$$\frac{1}{(x+3)^{4}}>10,000 \Leftrightarrow(x+3)^{4}<\frac{1}{10,000} \Leftrightarrow|x+3|<\frac{1}{\sqrt[4]{10,000}} \Leftrightarrow|x-(-3)|<\frac{1}{10}$$

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Video Transcript

this problem forty one of this tour calculus In addition, section two point four How close to negative three Do we have to take X so that one divided by the quantity X plus three to the fourth power is greater than ten thousand? First we'Ll solve this algebraic lee we'LL divide by ten thousand on about science while also multiplying by the quantity Extra story to thie fourth power They won't take the forth route on both sides Here we'LL have X plus three that quantity to the fourth to the one fourth power Right, So we're doing one fourth power to vote science so that we can cancel out forthe here on the left We should be really able to reduce system into one tent because ten to the fourth power is ten thousand on the red side. We want to assure that this quantity as illustrious positive So we write it as the absolute value of X mystery. This specifically is the distance between value's X and negative three. And when we see that it must be values Liston one ten toe or point one And so that's how close we have to be and we can also do this graphically. Take this function one over the quantity X plus three to the fourth power and plotted against ten thousand to see where this across the function. And we see in this case that our heartbreak solution is correct within a point one anything less guarantees a value greater than ten thousand on exactly as zero point one is where I won't get the exact value of ten thousand. So the answer is that we have to be closer than zero point one in order for the statement to be true.