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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 ...

0
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0answers
14 views

Prove that $\frac {2n-1}{2} - \frac {2n-2}{3} + … - \frac 2{2n-1} + \frac 1{2n}=\frac1{n+1} + \frac3{n+2}+…+\frac{2n-1}{2n}$

Prove that $\frac {2n-1}{2} - \frac {2n-2}{3} + ... - \frac 2{2n-1} + \frac 1{2n}=\frac1{n+1} + \frac3{n+2}+...+\frac{2n-1}{2n}$ for all $n \in \mathbb{Z^+}$ This question for me was pretty confusing ...
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3answers
26 views
2
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3answers
41 views

Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$

Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $$\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$$ I have attempted multiple times in this question and the only method that I ...
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2answers
26 views

Polynomial division problem- find the degree of the remainder

Let $r(x)$ be the remainder when the polynomial $x^{135}+x^{125}-x^{115}+x^5+1$ is divided by $x^3-x$. Then a. $r(x)$ is the zero polynomial b. $r(x)$ is a nonzero constant c. the degree of $r(x)$ is ...
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3answers
37 views

Let $a$, $b$, $c$, $d$ be positive real numbers. Prove that $(a^2 + b^2 )(c^2 + d^2 ) ≥ (ac + bd)^2$ [on hold]

Let $a$, $b$, $c$, $d$ be positive real numbers. Prove that $(a^2 + b^2)(c^2 + d^2) ≥ (ac + bd)^2$.
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3answers
62 views

4 simultaneous equations in real numbers

Solve the following system of equations in real numbers: $\begin{cases}x^2+zx=y+z\\y^2+xy=z+x\\z^2+yz=x+y\\xyz=1\end{cases}$
2
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1answer
46 views

Common fraction for :$\frac{1}{x^2-2x+2}+\frac{2}{ x^2-2x+3}=\frac{6}{ x^2-2x+4}$

Hi guys please help me with this equation: $$\frac{1}{x^2-2x+2}+\frac{2}{ x^2-2x+3}=\frac{6}{ x^2-2x+4}$$ My problem is with finding common fraction for denominators ($x^2-2x+2$ and $x^2-2x+3$and $x^...
0
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0answers
16 views

Formula : I'd like to multiply 2 numbers and modify the result to skew higher when larger numbers are involved and lower with lower numbers.

I'm not sure of the correct terminology here so I apologize if I've missed existing answers but when I searched I didn't find what I was looking for so I hope you can help. Here is the situation. We ...
2
votes
4answers
104 views

Rewrite $f(x,y) = 1-x^2y^2$ as a product $g(x) \cdot h(y)$

Rewrite $f(x,y) = 1-x^2y^2$ as a product $g(x) \cdot h(y)$ (both arbitrary functions) To make more clear what I'm talking about I will give a example. Rewrite $f(x,y) = 1+x-y-xy$ as $g(x)h(y)$ If ...
1
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2answers
45 views

Why does the interval [1,1) not contain 1?

On a quiz I was asked whether the interval [1,1) contained the number 1, and I answered that it was true, but apparently the answer is false. But why doesn't it? I thought that "[1,1)" means "all ...
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votes
2answers
47 views

Proving whether or not there exist $a,b$ such that $a+b>100$ and $a^2+b^2<1/1000$

sorry for my English unfortunately I don't know English so well Prove if the statement is true or false: For two real positive numbers a,b: a+b>100 and a^2+b^2<1/1000 I know the answer is false ...
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2answers
56 views

How to solve $(x^2-4x+4)^2=0$? [on hold]

How to solve this equation ? $$(x^2-4x+4)^2=0$$
1
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2answers
62 views

How can the following problem solved?

If Fatima sells 60 identical toys at a 40% discount on the printed price, then she makes 20% profit. Ten of these toys are destroyed in fire. While selling the rest, how much discount should be given ...
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1answer
27 views

When do I apply the distributive property?

I'm a bit lost here.... Equation 1: $(5p − 6) + (1 − p)$. Shouldn't I apply distributive property here? By distributing the '$+$' sign into $(1 - p)$ to give $(1 + p)$? If that is the case, then the ...
2
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0answers
29 views

What's the general equation of a 3D right cone? And the intersection of the 3d cone with a line?

I need to graph a cone about an arbitrary axis $Z'$ and a vertex with position vector $V$. My cone has equation: Equation of the Cone Now I need to find the intersections between this cone and a ...
0
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0answers
38 views

“Sort these numbers…”. Ascending, descending or how? [on hold]

I have the following exercise: Sort these numbers: $7$, $5$, $10$, $-100$, $\sqrt{7}$, $2.6$, $\log_3{7}$ By sort, what should I understand? Ascending, descending ...
1
vote
1answer
35 views

Find out the interval. [on hold]

Let $a,b,c, d$ be real numbers such that $a+b+c+d=0$ and $a^2+b^2+c^2+d^2=1$. Then the smallest possible value of expression $(a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2$ lies in the interval: i)$(0-1.5)$ ii)$(...
1
vote
1answer
10 views

Injectivity parameter

For a function $f:(-∞; 1]$ $\rightarrow$ ${R}$ $$f(x)=x^2+mx+1$$ Determine $m$ for which $f$ is injective(~stricly monotonous), without use of calculus. Based on my attempts, I've already restrained ...
0
votes
0answers
19 views

Explain why in a ring the additive neutral can not have a multiplicative inverse. [duplicate]

Explain why in a ring the additive neutral can not have a multiplicative inverse.
1
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3answers
81 views

How to prove $\sqrt{2} + \sqrt[3]{2}$ is an algebraic number

I need to show that this is an algebraic number by showing that it is a solution to: $x^{6} - 9x^{4} - 4x^{3} + 27x^{2} + 36x -23 = 0.$ I'm really struggling with this one, I tried squaring and ...
2
votes
1answer
46 views

Verify proof that $x_n = \sum_{k=1}^n {k \over 2^k}$ is bounded and find its supremum and infinum

The problem I'm solving states: Let $n\in \mathbb N$ and $x_n$ be a sequence: $$ x_n = \sum_{k=1}^n {k \over 2^k} $$ Prove $x_n$ is bounded and find $\sup\{x_n\}$ and $\inf\{x_n\}$ Let $S_n$ ...
0
votes
3answers
49 views

How should I interpret these trigonometry instructions?

What is the difference between these two instructions? I can do the first one, but I am not sure how to do the second. Without using a calculator, evaluate the following trigonometric functions for $\...
3
votes
5answers
47 views

Injectivity proof

Prove the injectivity(~strict monotony) of the following function, without utilising calculus: $f:(0;∞)$ $\rightarrow$$(0;1]$ $f(x) = \sqrt{x + 1} - \sqrt{x} $ I have already determined the given ...
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votes
1answer
53 views

How to prove the infinitude of twin primes? [on hold]

How to prove the infinitude of twin primes? Prove it by using some identity or some algebraic equations or means.
1
vote
1answer
80 views

For $a,b,c,d\in \Bbb N$ prove that $(a-b)(c-a)(d-a)(c-b)(d-b)(d-c)$ is divisible by $12$

Given that $a,b,c,d\in \Bbb N$ and pairwise distinct, prove that the number $$(a-b)(c-a)(d-a)(c-b)(d-b)(d-c)\tag{1}$$ is divisible by $12$. Source: IME 1999/2000 entrance exam My attempt: ...
3
votes
3answers
66 views

Domain and range of $g(x) = \frac{3x^2+4}{x-2}$

$$g(x) = \frac{3x^2+4}{x-2}$$ for the domain, I find, $\{x|x\ne 2\}$' and for the range $y = \dfrac{3x^2+4}{x-2}$ and when $x=2-\epsilon$, $y \to \dfrac{3(2-\epsilon)^2 + 4}{(2-\epsilon)-2} = \...
3
votes
2answers
40 views

If $x=\frac{(r-p)(s-q)}{(r-q)(s-p)}$, what is $\frac{(s-r)(q-p)}{(q-r)(s-p)}$ in terms of x?

If $x=\frac{(r-p)(s-q)}{(r-q)(s-p)}$, $y=\frac{(s-r)(q-p)}{(q-r)(s-p)}$ I've been asked to rewrite y in terms of x. I've tried guessing the solution and it is: $1-x=y$ Is there a more concrete way ...
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1answer
58 views

Are there number systems that fix divide-by-zero? [duplicate]

Natural numbers are closed under addition and multiplication, but not subtraction. Fixed by... Integers are closed under subtraction, but not division. Fixed by... Rational numbers are closed under ...
0
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3answers
47 views

For which values of $x$ does $y = \log_e (x - 4)$ become negative?

The logarithmic function is not defined for any value less then or equal to zero. Here, if $y=\log_e(x−4)$ is logarithmic function then its domain is $x \in (4, +\infty)$, and range is what we get ...
1
vote
1answer
38 views

Find exhaustive range of $k$ such that $f(x)=\frac{x-1}{k-x^2}$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$

Find exhaustive range of $k$ such that $$f(x)=\frac{x-1}{k-x^2}$$ never belongs to $\left[-1 \:\: \frac{-1}{3}\right]$ My try: Letting $$y=\frac{x-1}{k-x^2}$$ we get $$yx^2+x-(1+ky)=0$$ and since $...
1
vote
4answers
741 views

Wrong solution for solving $ 3x^2 - 6x - 9 = 0 $ [on hold]

Very Easy question, but I wasn't actively doing the question and I got it wrong. I was looking at it & didn't know which step was wrong? It looked like all the individual steps are correct but the ...
0
votes
1answer
19 views

Percentage of Outcome

I have a vendor bill which I need to pay. It is for $800. The government mandates that before I pay the bill I deduct 2.73% from the paid amount of this bill because the vendor is non compliant. How ...
1
vote
1answer
24 views

Complex Recursion

Consider the recursive function $C_n = C_{n-1} + iC_{n-2}$, where $C_1 = 1, C_2 = 1.$ If $C_{10}$ is written in the form $a+bi,$ find $b$. I solved this problem through brute force with a ...
2
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3answers
51 views

Check proof that $\prod_{k=1}^n(1+{1\over a_k})$ is bounded if $a_{n+1} = (n+1)(a_n + 1)$ and $a_1 = 1$, $n\in \mathbb N$

Let $n \in \mathbb N$ and: $$ \begin{cases} a_1 = 1 \\ a_{n+1} = (n+1)(a_n + 1) \end{cases} $$ Prove that $$ x_n = \prod_{k=1}^n\left(1+{1\over a_k}\right) $$ is a bounded sequence. Obviously ...
1
vote
1answer
42 views

Is that how they did it? (Solving integrals by substitution solution)

In case you're wondering, this part is taken from the answer key of my textbook, and was in the midst of an integrating by substitution solution, hence the u, (I just didnt write the whole thing out) ...
0
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1answer
23 views

Given piecewise-defined functions $f$ and $g$, for which $x$ is $f(x) > g(x)$?

This is a homework question from a precalculus class that I'm a TA for. Define the functions $$ f(x) = \begin{cases} 2x+12 &\text{if }\; x<0 \\ 5 &\text{if }\; 0\leq x \leq 6 \\ ...
1
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2answers
19 views

Proving that changing signs inside absolute value is valid for any two real numbers: $\forall x,y\in\mathbb{R}[\rvert x-y \lvert = \rvert y-x \lvert]$

I am just beginning learning about proofs, which is pretty cool, and I wanted to prove that $\forall x,y\in\mathbb{R}[\rvert x-y \lvert = \rvert y-x \lvert]$. I was wondering if my proof is correct: ...
-3
votes
2answers
63 views

Solving elementary school mathematics problem using best approach [on hold]

I have a mathematics problem : $$\sqrt[4]{\frac25} - \sqrt{\frac9{10}} - \sqrt{\frac52}$$ It is a multiple choice question with possible answers $0$, $\sqrt{\frac{5}{2}}$, $\frac{1}{\sqrt{10}}$ or $\...
0
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2answers
39 views

How to simplify this indices question? $\frac{3^n -3^{n-1}}{3^n / 3^{n-3}}$

Hi i got this indices question here $\frac{3^n -3^{n-1}}{3^n / 3^{n-3}}$ , I am not quite sure how to proceed and here's my workings $\frac{2 * -3^{n-1}}{1/{-3^3}}$ So far I can only do up to here, ...
1
vote
1answer
24 views

Graphical description of $g(x)=f(1/x)$?

I am working on a problem set out of Spivak's Calculus, and I am stuck on the following problem: Describe the graph of $g(x)$ in terms of the graph of $f(x)$. $$g(x)=f\left(\frac{1}{x}\right)$$ ...
0
votes
0answers
19 views

match each differential equation with its direction field [on hold]

match each differential equation with its direction field.i stuck this problem and don't know how to solve
0
votes
2answers
48 views

Find the intersection of two functions, $f(x) = \tan x$ and $g(x) = 5x$.

I can't proceed with the normal method to find these ones because they will intersect infinite times. Should I take an interval? $$f(x)=\tan(x), \qquad g(x)=5x$$ Thank you.
12
votes
4answers
161 views

$x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$ is rational or irrational?

The number $x$ defined below is rational or irrational? $$x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$$ From: IMO 1973 - Longlist My attempt (my real question is at the end): the identity $a^3+...
0
votes
0answers
20 views

Completing the square to find the Fourier transform of a Gaussian [duplicate]

I'm trying to find the Fourier Transform of a Gaussian, and I end up having to complete the square in the argument of an exponential so I can use the standard Gaussian integral. I basically have: $$\...
1
vote
2answers
31 views

Evaluate $\int_{3}^{10} \left[ \log \left[x\right]\right]dx$

Evaluate $$I=\int_{3}^{10} \left[ \log \left[x\right]\right]dx$$ where $[.]$ is Greatest integer function. My try: we have $$I=\sum_{r=3}^{9}\int_{r}^{r+1}\left[\log r\right]dx$$ but how can we ...
2
votes
3answers
76 views

Does $\sqrt{i^4} = i^2$?

I'm assuming it doesn't, because if it did, then $1 = \sqrt{1} = \sqrt{i^4} = i^2 = -1$. In general, does $\sqrt{x^4} = x^2$?
0
votes
0answers
34 views

Probability of $P1$ winning championship [duplicate]

Two players $P_1$ and $P_2$ are playing the final of a chess championship,which consists of a series of matches.Probability of $P_1$ winning a match is $\frac{2}{3}$ and that of $P_2$ is $\frac{1}{3}$....
0
votes
1answer
19 views

Single or multiple solution of linear equation

For the two linear equations: \begin{align} x + xy^2 &= 40y\\ x - xy^2 &= -32y \end{align} I find that one solution for $(x,y)$ is $(12, 3)$. How can I determine if there are any other ...
0
votes
1answer
20 views

resistor values in non-inverting op amp with bias

$$ \newcommand{\rth}{R_{TH}} \newcommand{\vth}{V_{TH}} \newcommand{\rf}{R_{F}} $$ I'm stumped (hence posting here!) with what is probably a very simple extrapolation. Consider this circuit: The ...
0
votes
4answers
28 views

Domain and range of $g(x) = 3 /(x-4)$

$$g(x) = \frac{3}{x-4}$$ for the domain, I find, $\{x|x\ne 4\}$' and for the range $$\{y|- \infty < y < 0 < y < \infty \}$$ $$\{y| 0 < |y| < \infty \}$$ Firstly, have I got that ...