# 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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### Explaining the fence-post formula [duplicate]

From the book: Set Theory for Begginners I call the formula “n-m+1” the fence post formula . If you construct a 3-foot fence by placing a fence-post every foot ,then the fence will consist of 4 fence ...
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### Prove that $\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}$

If $a+b+c=3$ ,$a,b,c>0$ and $\lambda \geq 1$, prove that : $$\frac{a}{a^2+\lambda}+\frac{b}{b^2+\lambda}+\frac{c}{c^2+\lambda} \leq \frac{3}{\lambda +1}.$$ my attempt: using CBC twice and ...
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1 vote
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### Why do you have to check all the 'sections' of an inequality when solving for x?

For instance, the inequality $x^2+3x+2>0$ factors into $(x+2)(x+1)>0$ If this were just an equation (i.e ...=0) you would know the solutions are x={-2, -1}. But, because it's an inequality, you ...
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### If the ratio of principle and the monthly simple interest is 192:1. What is the yearly rate of simple interest? [closed]

Please help me If the ratio of principle and the monthly simple interest is 192:1. How can I find the yearly rate of simple interest?
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### Computing the log of a sum of exponentials

in a Coursera course by UW I've come across this piece of code computing the log of a sum of exponentials. ...
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### Let $a_n$ be the number of $X$-strings of length $n$ that do not contain $344$ as a sub-string. Find a recurrence relation for $a_n$.

Let $X = \{1,2,3,4\}$ and $a_n$ be the number of $X$-strings of length $n$ that do not contain $344$ as a sub-string. Find a recurrence relation for $a_n$. We can construct a string of length $n$ in ...
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### Can we draw a closed path made up of 9 line segments , each of which intersects exactly one of the other segments?

The solution given in Fomin's book is as follows. If such a closed path were possible, then all the line segments could be partitioned into pairs of intersecting segments. But then the number of ...
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### $365(x)^{364}-365(x)^{365}+x^{365}=0.9$

I can't seem to solve this, I tried using multiple software but it says it doesn't support this kind of equation: $$365(x)^{364}-365(x)^{365}+x^{365}=0.9$$ Context: Hey! I was having fun with tricked ...
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### match between a function graph and it's derivative graph [closed]

I need help with this question I only manage to match 3 functions: 1-B 2-C 3-D
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### Is $0$ the root of the equation $\frac{x^2}{ x}= 0$?

I want to understand if I understand the concept of equation correctly. For this I want to know if $0$ is the root of the equation $\frac{x^2}{x} = 0$. If we simplify the equation then we can get ...
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### Eliminating x = 0 from an equation when x can be equal to all real numbers

If we have the equation: $$2+0x=a+0x$$ The solution to this equation for $x$ is all real numbers. If I then multiply each side by $x$ I get: $$2x=ax$$ The solution to this equation for $x$ is still ...
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### Why is the Arccosine of $30$ degrees undefined?

Recently, while working on Trigonometry, a problem came up in which I was asked to evaluate the value of $\cos(\arccos(30^\circ))$, and I stated that the value of this function was $30$ degrees (...
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### Train Traveling from Aytown to Beetown Accident (AHSME 1955)

I was working on the question: A train traveling from Aytown to Beetown meets with an accident after 1 hour. The train is stopped for 30 minutes, after which it proceeds at four-fifths of its usual ...
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### fence-post formula [closed]

Would you explain this fence-post formula : n-m+1 exactly ,why is +1?
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### Why am I getting extraneous solutions solving $\frac{1+\sqrt{1-x}}{x-\sqrt{1-x^{2}}}=2 x$?

I came across a beautiful problem: Solve $$\frac{1+\sqrt{1-x}}{x-\sqrt{1-x^{2}}}=2 x$$ My approach: Obviously $x=0$ is not a solution. Now we have: $$1+\sqrt{1-x}=2x^2-2x\sqrt{1-x^2} \tag1$$ ...
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### Extracting the leading non zero element from a list

Given a circular list of 10 numbers. Each element $e_i$ in the list ($0 \leq e_i \leq1)$. There are always either 1 or 2 non zero elements in the list. Case 1: In case we have 1 non zero element, it ...
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### Queuing models - Solving problem with different service time 1/μ

I'm struggling with a problem because the service time is different depending on the number of tickets purchased. hereinafter the case: "The owner wants to reduce the waiting and serving time for ...
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### What is the proof for the $(\sqrt[n]{a})^n$ equaling $a$ or $|a|$ if $n$ is odd or even?

If I use $64$ and $-64$ as the radicands, and have $2$ or $3$ as the indices, I know that it all works out arithmetically (except for $\sqrt[2]{-64}$ which has no real solutions) but I'm not sure how ...
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### Finding the set of parameters for which the inequality holds for all $x,y$

I encountered the following question about inequalities which I am curious how to solve. The simplest case is to consider the inequality $$|x|+|y|+|x+y|+ax+by\geq 0$$ where $x,y,a,b\in\mathbb{R}$. The ...
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### Cities $A$ and $B$ are $999$km apart. Road signs every km show distances to each city. How many signs use only two digits (without trial and error)?

Recently, I came across a question that seemed as if it can only be solved using trial error. Here it is: There are two cities, $A$ and $B$, their distance is $999$km. Along a road that connects $A$ ...
1 vote
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### finding a linear map over natural numbers

Consider a map $f$ from $\mathbb{N}^N$ to $\mathbb{Z}$ as follows: $$f(n_1, n_2, \cdots, n_N) = n_1 m_1 + n_2 m_2 + \cdots + n_N m_N.$$ Where $n_i$'s are natural numbers with $0\leq n_i \leq M$. The ...
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### Sketching $X\cup Y$, $X\cap Y$, $X-Y$, $Y-X$, where $X=\{(x,y):x^2+y^2\leq1\}$ and $Y=\{(x,y):-1\leq y\leq 0\}$

Image contains the question as well as its answer. Expecting someone to verify the sketch.
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### How can you solve algebraic equations (eg, $10x+y=3xy$, knowing that $x$ and $y$ are integers between $1$ and $9$) without trial and error?

Often, I will come across questions which involve equations that need to be solved by trial and error. For example, the equation $$10x+y=3xy$$ In this case, the question provides us that $x\neq y$ and ...
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### Range of rational function in different ways.

I was looking for range of a rational function, I used three ways to get the answer and all of them are giving different results . e.g. f(x)=1/(1-x²) then, (1) ...
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