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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 ...

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0answers
7 views

Systems of Linear Equations: Antifreeze Drain and Replace Problem

The radiator in your car contains 4 gallons of antifreeze and water. the mixture is 45% antifreeze. How much of the mixture should be drained and replaced with pure antifreeze in order to have a 60% ...
-1
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1answer
25 views

Question about trace of matrix

In literature, I found the following identity. Unfortunately, I fail to see why this holds. Where does the trace of the matrix come from? Any clarifications are highly appreciated!
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1answer
9 views

Bivariate transformation of random variables: brute force algebra?

Suppose I have 2 random variables, $Z_1$ and $Z_2$. I then define the following bivariate transformations, \begin{equation} X = a_xZ_1 + b_xZ_2 + c_x \end{equation} $$Y = a_yZ_1 + b_yZ_2 + c_y$$ ...
1
vote
4answers
47 views

Finding range of $\sin(x)\cos(2x)$

I have to find the range of $f(x)=\sin(x)\cos(2x)$. Here's what I have so far: $f'(x)=-\cos(x)\cos(2x)-2\sin(2x)\sin(x)$. So $\cos(x)\cos(2x)+2\sin(2x)\sin(x)=\cos(2\cos^2(x)-1)+4\sin^2x\cos x=0$. ...
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votes
2answers
42 views

Solving $2y=\sqrt{3+\frac{1}{2y}}$

Any way to solve this irrational equation in $\mathbb{R}$? I think it has some artifice, but I do not see it $$2y=\sqrt{3+\frac{1}{2y}}$$
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1answer
17 views

Range of exponential functions

I have to find the range of $4^{\sin(x)}+ 2^{\sin(x)+3}=2^{2\sin(x)}+ 8\cdot2^{\sin(x)}$. Let's take $y=2^{\sin x}$, so we rewrite the equation as $y^2+8y=0$. The range of this function is $[-16,+\...
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2answers
36 views

Find the range of $f(x)=\log_3(x) +log_x(3)$

I have to find the domain of the function $f(x)=\log_3(x) +log_x(3)$. I know that for some functions we differentiate it and check the function in critical points. However, differentiated this but ...
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2answers
38 views

Given a triangle with coners $A,B,C$, Show $\sin(A)+\sin(B)+\sin(C) \leq \frac{3\sqrt{3}}{2}$

Given a triangle with vertices $A,B,C$, Show $\sin(A)+\sin(B)+\sin(C) \leq \frac{3 \sqrt{3}}{2}$. Here is a proof using Jensen inequality: $\sin(x)$ is concave from $0$ to $\pi$. hence $\frac{\sin(A)...
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2answers
52 views

Find the value of $f(g(x))$ [on hold]

If $$f(x)=-1 \hspace{0.2cm} \forall\hspace{0.2cm} x<0 \\=0 \hspace{0.2cm} \forall \hspace{0.2cm}x=0\\ \hspace{0.2cm} =1\hspace{0.2cm} \forall \hspace{0.2cm} x>0$$ respectively, and $$ g(x)=1+...
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3answers
36 views

Unclear what steps to take…

I am trying to simplify $$\frac{6^{\frac3{12}}} {3^{\frac 4{12}}}$$ which should be $$\frac {2^{\frac 14}}{3^{\frac 1{12}}}$$ But I fail to see what step to take to get there... I thought we couldn't ...
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1answer
10 views

What conditions must be present in a word problem to set it up and solve as a system of linear equations (2 variables)?

I was solving various word problems such as mixing solutions, differing interest rates, boat speeds, etc., and I thought what's the common underlying factor among these problems which allow us to ...
3
votes
1answer
56 views

Irrational equation $\sqrt{9-4x}=p-2x$

The equation $$\sqrt{9-4x}=p-2x$$ has exactly 2 real and different solutions only if parameter $p$ belongs to which set? So what I see here, to have the solutions be real in the first place, $9-...
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0answers
32 views

Theory of equation [on hold]

$\begin{align} & {{x}^{2}}+{{y}^{2}}=1+\frac{2xy}{x-y} \\ & \\ \end{align}$ $\sqrt{x-y}={{x}^{2}}+5y$ What will be the maximum value of $xy$ ?
1
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1answer
28 views

Prove that $C_{3 \over 2}^n$ is bounded given $C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}$

Let: $$ \begin{cases} C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}\\ C_{a}^0 = 1 \end{cases} $$ Prove $C_{3 \over 2}^n$ is bounded. I've started with finding a reduced formula: $$ C_{3\over 2}^n ...
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1answer
26 views

Multiplying prices with one in cent is the same as them all added

Is it possible for all prices out of a list of prices (in Euro/Dollar) multiplied together, with one of them in cent however, to amount to the same total of 20.18€ as all of them added together? (...
3
votes
1answer
33 views

Show that $|y_{n+1} - x_{n+1}| \le \frac{|b-a|}{4^n}$ for $x_{n+1} ={1\over 2}(x_n + y_n)$ and $y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)}$

This question is the last part of the problem statement which was not included in this question. Let: $$ \begin{cases} x_{n+1} = {1\over 2}(x_n+y_n)\\ y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)} \\...
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0answers
38 views

How to prove the Inequality expression

is it possible to prove the two expression : $if (\ 0\le\ a\le1\ \ ,\ 0\le\ b\le1,\ 0\le\ c\le1)$ $ ,(\ a\le\ b\le\ c\ )\ ,\ and\ (\ a+b+c=1)$ Then : $ a^2+b^2-ab\le\ a^2+c^2-ac$ $if (\ 0\le\ ...
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3answers
29 views

How do i do this mathematical induction question?

My question:$5+10+20+...+5(2)^{n-1} = 5(2^n -1)$ So first step i have to prove LHS = RHS when $n=1$, which is true. Then I assume the statement is true for $n=k$ Since the statement is true for $n=k$ ...
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3answers
50 views

Formula, making the subject.

Make a the subject, $$\sqrt{c-a^2}=a+b\tag1$$ $$c-a^2=(a+b)^2\tag2$$ $$c-a^2=a^2+b^2\tag3$$ $$c-b^2=a^2+a^2\tag4$$ $$a^2=\frac{c-b^2}{2}\tag5$$ $$a=\sqrt{\frac{c-b^2}{2}}\tag6$$ My teacher said ...
1
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2answers
39 views

Is $\sqrt{x}$ an even function

I'm going through some pre-calculus and I am presented with this rule. If $f(x)$ is an even function, then $f(x) = f(-x)$ So with the following example I have: $$f(x) = 3x^4 \\ f(-x) = 3(-x)^4 ...
1
vote
3answers
49 views

$x$ intercept problem

How would I find the $x$ intercept of $x^5-x^3+2=0$? I haven’t learned about things like synthetic division or any theorems, just algebraic manipulations.
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1answer
22 views

There is a general method for finding ranges of function without resorting to calculus?

I'm having trouble to find the range of more complicated functions such as $ f(x) = \frac{1}{\sqrt{x + 1}} $ How one should proceed to tackled down these functions, specially the ones involving ...
0
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4answers
39 views

Can you please explain what's wrong with the following reasoning?

Suppose we want to solve the following inequality $ \frac{1}{x} + 1 >0 $ So we proceed as follows $ \frac{1}{x} > -1 $ $ x < -1 $ Thus $ x \in (-\infty, -1)$ But the solution to the ...
1
vote
4answers
46 views

Will this inequality stay preserved if I take $\log$ on both sides? EDIT: Also, can someone please try and help me work towards a solution?

I am trying to solved this inequality for $k$. $x^{2k}<\varepsilon\cdot k^k$ Here $k\in\mathbb{N}$ and $x,\varepsilon$ are fixed such that $x,\varepsilon\in\mathbb{R}$ and $\varepsilon>0$. I ...
1
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1answer
40 views

Verify proofs related to monotonicity of $x_{n+1} = {1\over 2}(x_n+y_n)$ and $y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)}$

Let $\{x_n\}$ and $\{y_n\}$ be sequences defined by recurrence relations: $$ \begin{cases} x_{n+1} = {1\over 2}(x_n+y_n)\\ y_{n+1} = \sqrt{{1\over 2}(x_n^2 + y_n^2)} \\ x_1 = a > 0\\ y_1 = b > ...
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votes
4answers
56 views

How do I find the roots of $y=e^x-2$?

I need to find the root of this: $y=e^x-2$. I don't know how to find the roots of this function because I have two variables... The teacher said the root is between $0$ and $1$.
2
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3answers
53 views

Show that $x_{n+1} = \frac{2+x_n^2}{2x_n}$ is a decreasing sequence.

Let $x_n$ be defined as: $$ \begin{cases} x_{n+1} = \frac{2+x_n^2}{2x_n} \\ n\in \mathbb N \\ x_1 = 4 \end{cases} $$ Show that $x_n$ is a decreasing sequence. I'm having a hard time with the ...
17
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6answers
2k views

What are the possible values of these letters?

Out of all the questions I answered in a math reviewer, this one killed me (and 7 more). Let $J, K, L, M, N$ be five distinct positive integers such that $$ \frac{1}{J} + \frac{1}{K} + \frac{1}{L} + \...
17
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2answers
2k views

Can we make it equal to x?

$\sqrt{6 +\sqrt{6 +\sqrt{6 + \ldots}}}$. This is the famous question. I have to calculate it's value. I found somewhere to the solution to be putting this number equal to a variable $x$. That is, $\...
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3answers
61 views

New numbers discovered! Limit of $8/10$, $98/100$, $998/1000$, $\ldots$ is $1$, yet it's not, because of the $8$. [on hold]

So, I am starting out with $8/10$ (or $a/10$, for that matter). Proceed to form $98/100$, then $998/1000$, then again $9998/10000$ and so on up to: $$\frac{\ldots 9998}{10000\ldots}$$ Clearly, the ...
1
vote
2answers
53 views

Function such that $f(\frac{1}{n}) = n/(n + 1)$

I want to prove something about functions that satisfy $$f\left(\frac{1}{n}\right) = \frac{n}{n + 1}.$$ Obviously, one thing that we can do is $$f(x) = \frac{1}{1 + x} $$ because we can write $\...
2
votes
2answers
57 views

How to solve $1/x = 1/a + 1/b$ to get $x = ab/(a + b)$?

I'm working through the first problem set in a text book. I have the question and the solution, however the solution gives no in between steps, only the final result. The question is solve for $x$ in ...
2
votes
0answers
48 views

Combinatorial Function Factoring

Define ${x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!}$ for positive integer $n$. For what values of positive integers $n$ and $m$ is $g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}}...
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1answer
20 views

Probability for *given* questions

I have a question for probability. In probability there something called a given which can be represented by | which I will use in this submission. For example the question: P(B|A) would utilise the ...
0
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1answer
41 views

common point on the three lines…

Actually, I am dealing with a problem in barycentric coordinates I got the equations of three lines as I know that these three lines sharing a common point, I know if I prove their det is zero, they ...
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3answers
25 views

Trigonometry Addition and Subtraction Formula

My professor showed us how to solve $\cos(\theta - X)$ where $\cos(\theta) = \frac{3}{5}$ and is in Quadrant IV, and $\tan(X) = -\sqrt{3}$ and is in Quadrant II. Since, $\cos(A-B) = \cos(A)\cos(B)+\...
2
votes
3answers
51 views

challenge trig question - no calculator

The challenge trignometry question is: simplify $sin (80^\circ) + sin (40^\circ) $ using trignometric identities. All the angle values are in degrees. This is what I did: Let $a=40^\circ$. So we ...
5
votes
2answers
70 views

Stuck on $6y-3-(2y-1)=12$

Solve for y: $$6y-3-(2y-1)=12\tag1$$ $$6y-3-2y-1=12\tag2$$ $$4y-4=12\tag3$$ $$4y=12+4\tag4$$ $$y=4\tag5$$ But when I plugged into $(1)$ it is wrong, $y=4$ $$6y-3-(2y-1)=12\tag1$$ $$24-3-(8-1)=...
0
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1answer
21 views

Using distributive property to prove that the product of two odd numbers is odd

I've been reading Algebra The Easy Way, and there's a problem at the end of the chapter to prove that the product of two odd numbers is an odd number. In this problem, an even number is defined as ...
0
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4answers
150 views

Let $f(x)=x^3 - 3x + 1$ [on hold]

Let $f(x)=x^3 - 3x + 1$. a) Expand $(x+h)^3$. Would this simply be $x^3 + h^3$? b. Use the formula $f’(x) =\lim_{h\to 0}\big(f(x+h)-f(x)\big)/h$ to show that the derivative of $f(x)$ is $3x^2-...
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2answers
20 views

Is this solution for a equation above 2nd degree correct?

I have a simple equation above 2nd degree that I solved, with that I mean that my solution matches the one on the book, but I wonder if my method is correct. $2x^5-32x=0 $ I factor out 2x $2x(x^4-...
-1
votes
2answers
53 views

Inverse of $\ln(e^x-3)$

so the whole concept about inverses is a little foggy. Say you have function $f(x)=\ln(e^x-3)$ and you want to know the inverse function, then: $$\ln(e^x-3) = y$$ $$e^x-3 = e^y$$ $$e^x = e^...
2
votes
5answers
46 views

Show $x_{n+1} = {1\over 2}x_n^2 - 1$ is bounded below and unbounded above and $x_n$ is increasing.

Let: $$ \begin{cases} x_{n+1} = {1\over 2}x_n^2 - 1\\ x_1 = 3\\ n\in \mathbb N \end{cases} $$ Show that the sequence $x_n$ is bounded only below and is increasing. I've started with the ...
-1
votes
2answers
36 views

What is the difference between the summations?

What is the difference between the summation $$\sum_{1 \leq i<j \leq n} f(i,j)$$ and $$\sum_{1\leq i} \sum_{<j \leq n} f(i,j)?$$
3
votes
1answer
40 views

Show that $x_{n+1} = \frac{x_n + 1}{n+1}$ is decreasing starting from some $n_0$ and find $n_0$.

Given $n\in \mathbb N$ and: $$ \begin{cases} x_{n+1} = \frac{x_n + 1}{n+1} \\ x_1 = -10 \end{cases} $$ Show that $x_n$ is decreasing starting from some index $n_0$. Find $n_0$. I've tried to ...
-1
votes
1answer
36 views

Proving $(1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d$ [duplicate]

Prove the following inequality for $a,b,c,d \in (0,1)$: $$(1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d$$ I have a problem. I don't know if my idea is good $a=b=c=d $ $(1-a)^4 > 1- 4a $ So, this is ...
1
vote
1answer
25 views

Binomial Expansion rth term

Given that there's no term on $x^3$ in $$(k + 2x) (1 - 3/2 x) ^6.$$ Evaluate the value of $k$ Difficult After expanding $(1 - 3/2 x)^6$ up to the 3rd term I get $1 - 9x + \frac{135}{2} x^2 - \frac{...
-1
votes
1answer
24 views

find equal 2nd and 3rd dimensions of rectangular prism when given volume and one side [on hold]

I am interested to know how to calculate the length and width (which should be equal) of a rectangular prism knowing only the overall volume and the height. So for example ...
0
votes
1answer
38 views

If $a-b=3,\ a+b+x=2,$ then the value of $(a-b)[x^3+3(a+b)x^2+3x(a+b)^2+(a+b)^3]$

Question: If $a-b=3,\ a+b+x=2,$ then the value of $\left(a-b\right)\left[x^3+3\left(a+b\right)x^2+3x\left(a+b\right)^2+\left(a+b\right)^3\right]$ is? I tried $a+b=2-x$ and $(a-b)(a+b) =a^2 +b^2$ = $(...
-2
votes
2answers
37 views

Equation with 3 unknown parameters? [on hold]

I am trying to solve this equation: ax + by + cz = d. Known parameters: a,b,c,d I also know that z has to be = x + y + 1 everytime. Unknown: x, y,z (=x+y+1) So for example a=2 b=3 c=4 d=20 would ...