# Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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### Solving an equation with a shift

My professor offered an extra exercise for us to think about. The problem is to solve the shifted equation of the form $$f(x+ia) = x^2 f(x) \, ,$$ where $a$ is a constant. Due to the $x$-dependent ...
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### What is the image of $\sin(1/x)\cdot(1/x)$, $0<x<1?$

What is the image of $\sin(1/x)\cdot(1/x)$, $0<x<1?$ I just had a question about what the image of $\sin(1/x)\cdot(1/x)$ is for $0<x<1$. Would it not be all the reals, since $\sin(1/x)$ ...
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### Proving $\frac{a+b+c}{a^2b+b^2c+c^2a+9}\ge \frac{abc+6}{abc+27}$ for $a, b, c\ge 0$; $ab + bc + ca = 3$

I came up with the inequality accidentally so there is no original proof so far. It would be great if you can give some useful help to prove it. Problem. Given non-negative real numbers $a,b,c$ ...
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### How to use factor theorem for a polynomial with three binomial factors, i.e. (a+b)(b+c)(a+c)

I am working on a proof and I am completely baffled on how to use factor theorem for more than one binomial factors (as divisors). I find that showing the LHS is equivalent to the RHS is more ...
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### Function problem, pre-university level

The function $f: R → [-10, +∞) / f(x) = \dfrac{14a-b}{a³(b+5)}x²+(2a-3b)x+(4a-b)$ is even and surjective. If $x_1$ and $x_2$ are their real roots, then $|x_1|+|x_2|$ is equal to? Answer: 9 I have been ...
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### i have a rosace curve with five petals (with r=1+4sin(5θ)), i want to calculate the area of colored part of the pétale using intégrales

This is an image of a rosace curve with five petals. One of the petals is colored in purple, and i do want find the area of the colored surface.
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### Is my solution to "MacPOW 1141: Capturing 5 integers" correct?

Crossposted from P.SE as it was closed because I sort of forgot the difference between a math puzzle and a math problem Source: MacPOW 1141 "MacPOW 1141: Capturing 5 integers" states: For ...
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### Problems on lcm and gcd.

Mario has a rest shift every $8$ days; Luigi every $24$ days; Paolo every $16$-th days. Today all three are off. In how many days will all three be back in rest shift for the first time? There are ...
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### Why is subtraction not associative if addition is? [closed]

We know that: $$\forall a,b,c \in \mathbb{R}: (a+b)+c=a+(b+c)$$ Subtraction can be defined as the addition of the additive inverse of a number. So $a-b-c$ can also be written as $a + (-b) + (-c)$ ...
1 vote
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### How do I ensure that I have not lost/gained any solutions when solving a trig/algebraic equation? [closed]

Sometimes when I try to solve an equation, I need to multiply by a $\cos(x)$ for example to create a common denominator. Does this create a new solution? Why does/doesn't it? When do I know if when I ...
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1 vote
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### How to algebraically calculate the difference in days for an inverse-proportion or more people=worse problem?

During a drought, 50 people have only $1000 \mathrm{~L}$ of water left. If every person consumes an identical amount of water, the 1000-liter supply would be exhausted in one day. If 40 people were to ...
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### Function to fit a set of data points [closed]

I am trying to write an equation to describe a line where the values would be as follows. I am so near yet so far. I seem to be unable to paste data from Excel without it coming in as a picture. I ...
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1 vote
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### Is there a short proof that $-(-n) = n$, when $n$ is a positive integer? [duplicate]

This maybe a silly question, but I was thinking about it. First we have the set $\mathbb{N}=\{0,1,2,3,...\}$ of natural numbers, then to create $\mathbb{Z}$ we do the following: For each natural ...
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### Find the minimum value of $\frac{1}{x - y} + \frac{1}{y - z} + \frac{1}{x - z}$ for real numbers $x > y > z$ given $(x - y)(y - z)(x - z) = 17$.
I've tried using AM-GM inequality: $$\frac1{x-y}+\frac1{y-z}+\frac1{x-z}\ge3\sqrt{\frac1{(x-y)(y-z)(x-z)}}$$ Which gives us $$\frac1{x-y}+\frac1{y-z}+\frac1{x-z}\ge\frac3{\sqrt{17}}$$ By ...