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Under what conditions is $\vert x-y \vert = \vert x \vert -\vert y \vert$?
This is a fairly basic question, however, I don't know what I am missing here. Solving the equation $\vert x- y\vert = \vert x \vert -\vert y \vert$ yields: \begin{align} \vert x- y\vert &= \vert ... algebra-precalculus absolute-value- 111
Advance polynomial forms
When the polynomial $f(x)$ is divided by $x^3 + x + 1,$ the remainder is 3. When the polynomial $f(x)$ is divided by $x^3 - x + 1,$ the remainder is $x + 1.$ Let $r(x)$ be the remainder when $f(x)$ is ... algebra-precalculus polynomials- 11
Number of solutions for $x ^ y = y ^ x = k$
For a given value of $k \geq 0$, how many solutions $x, y \in \mathbb R$ are there to $x ^ y = y ^ x = k$? My attempt so far: There is the "trivial" solution where $x = y$, and the problem ... algebra-precalculus exponentiation- 201
Polynomial factorization with three variables.
I need for factor the following polynomial. I have been looking for ways of doing it, but most of the resources show the simple example where everything beautifully factors. The problem is motivated ... algebra-precalculus- 71
how to prove the equation [closed]
vessel contains 2000 litres of sauce recipe which is pumped out of the vessel to be bottled at a rate of 60 litres/hr. Preservatives are added to the mixture at a rate of 80 litres/hr to which 8g of ... calculus algebra-precalculus derivatives maxima-minima- 1
Simplify $\sqrt[r-p]{(x)^{\frac{1}{r-q}}} \times \sqrt[q-r]{(x)^{\frac{1}{q-p}}} \times\sqrt[p-q]{(x)^{\frac{1}{p-r}}} $ [closed]
Simplify $\Large\sqrt[r-p]{(x)^{\frac{1}{r-q}}} \times \sqrt[q-r]{(x)^{\frac{1}{q-p}}} \times\sqrt[p-q]{(x)^{\frac{1}{p-r}}}.$ algebra-precalculus exponentiation radicals- 1
Solve $x^{2^{\sqrt{2}}} = {\sqrt{2}}^{2^x}$
How to solve: $$x^{2^{\sqrt{2}}} = {\sqrt{2}}^{2^x}$$ where $x \in R^{+}$? We take log based on 2 on both sides, then $2^{\sqrt{2}} \log_2 x = 2^x \log_2 {\sqrt{2}} = 2^{x-1}$ (thanks for the comment ... algebra-precalculus logarithms- 429
System of inequations
If $p$, $q$, $r$, $s$ and $t$ are real numbers such that $q+r<s+t$, $r+s<t+p$, $s+t<p+q$ and $p+q<r+s$, then find the largest and the smallest term among them. This is how I solved it: $$... algebra-precalculus inequality systems-of-equations- 339
How do we go from $x^n - a^n$ to $(x-a)(x^{n-1}+ ax^{n-2} +\ldots + xa^{n-2} + a^{n-1})$
This is involved in the proof of the standard limit x tends to a:$\dfrac{(x^n - a^n)}{(x - a)}$. How can we prove this statement and how do we know that the second term ends at $x^0$. Is this limit ... algebra-precalculus limits regular-expressions- 1
Inequality with discriminants
If $x^2-ax+1-2a^2>0$ for all $x \in {R}$, find range of $a$ The solution to this takes the discriminant of the expression in terms of $a$, i.e., $$\implies D={a^2-4(1-2a)}>0(\because x \in R)$$ ... algebra-precalculus inequality polynomials quadratics- 339
Relating $\sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}$ to $(\;\sum_{k=1}^N a_k e^{\frac{2\pi i}{N}k}\;)^2$
Consider the following expression $$ \sum_{k=1}^N a_k^2 e^{\frac{2\pi i}{N}k}\tag{1} $$ where $i$ is the imaginary number. How may I relate it to the following expression $$ \left(\sum_{k=1}^N a_k e^{\... algebra-precalculus summation fourier-transform sums-of-squares- 3,418
write the given summation in terms of $x^n$ instead of $x^{3n}$
I have $\sum_{n\geq0}(2n)x^{3n} =0+2x^3+4x^6+6x^9+...$ , but i want to write this summation in terms of $x^n$ instead of $x^{3n}$ .How can i do it ? I thought that if i can write $n/3$ in place of $n'... sequences-and-series algebra-precalculus summationThe limit of sequence in the recurrence relation form
If a sequence is defined by $u_{n+1} = k u_n + 9$ where $k$ is constant , if $u_1 = 5 $ and the limit of the sequence as $n\to \infty$ is $15$ .Find the value of $k$ . I know that i have to find the ... sequences-and-series algebra-precalculus- 1,734
How to calculate square root of 2 with this method? I'm not quite understandind ir from my college [closed]
Ilustraçao tirada do livro calculo difrencial e integral - piskunov calculus algebra-precalculus- 1
Arc length vs Surface of revolution.
I don't understand why these two problems are solved differently here the first one $fig(1)$ and 2nd one $fig(2)$. Why did we take the limit $\displaystyle \lim_{r\to0^+}\int_r^\pi \sqrt{2-2cost}\... calculus integration algebra-precalculus definite-integrals applications- 59
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