Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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10
votes
3answers
4k views

Why should you never divide both sides by a variable when solving an equation?

I'm currently working through an algebra book, and during the chapter about rational expressions and inequalities, the author has a side note in which he states: Never divide both sides of the ...
0
votes
1answer
541 views

How to calculate ratio of ppm (parts per million)?

If have a container with a capacity of 5,000 cubic inches, how do I calculate the ratio of 50 ppm? I would describe the problem as, 1.) statements, ...
8
votes
7answers
943 views

How would I solve $\frac{(n - 10)(n - 9)(n - 8)\times\ldots\times(n - 2)(n - 1)n}{11!} = 12376$ for some $n$ without brute forcing it?

Given this equation: $$ \frac{(n - 10)(n - 9)(n - 8)\times\ldots\times(n - 2)(n - 1)n}{11!} = 12376 $$ How would I find $n$? I already know the answer to this, all thanks toWolfram|Alpha, but ...
3
votes
5answers
979 views

What does the exclamation mark do?

I've seen this but never knew what it does. Can any one let me in on the details? Thanks.
3
votes
5answers
702 views

Finding diameter of the largest circle

What is the diameter of the largest circle that can be drawn on a chessboard so that its entire circumference gets covered by the black squares and no part of the circumference falls on any ...
2
votes
1answer
502 views

Prove that $\sup(S)=1$ if $S=\{x \in \mathbb{R}| x^2 < x\}$

I wanted to check whether I have done this proof right. I have not fully convinced myself. Proof: Let $S=\{x \in \mathbb{R}\mid x^2 < x\}$. Since $x \in \mathbb{R}$, we know that $x^2 > 0$, ...
3
votes
1answer
274 views

How to find a three digit number which,when reversed, becomes equal to $17$ times the square of it's cube root?

How to find a three digit number which,when reversed, becomes equal to $17$ times the square of it's cube root? If we assume that the three digit number is of the form $100x+10y+z$,where $x \in ...
0
votes
2answers
70 views

Mathematics functions

i have encountered myself with a mathematic question problem. The exercise says: The function $f$ is defined by $f:xa \sqrt{3-2x}$. Evaluate $f^{-1}(5)$. Does anyone have a clue how to resolve the ...
1
vote
1answer
117 views

How do I do this math right?

$(2 + 3x) * (4 + 5x) = 2(4 + 5x) + 3x(4 + 5x)$ I don't get it. What do I do?
0
votes
1answer
809 views

Multiplying exponents by fractional exponents and whole numbers to the power of fractional exponents

I am trying to understand the different rules for multiplying exponents by fractional exponents and raising whole numbers by the power of fractional exponents. I have an idea but I'm trying to assure ...
1
vote
1answer
219 views

Question about significant figures

I was taught that there are two different methods for obtaining results for multiplication/division or addition/subtraction with decimals. For multiplication/division the result will have the least ...
1
vote
1answer
503 views

Really simple question about sine wave equation with $\pi$

I'm reading some EE material, and my trigonometry is really rusty. There is an equation that looks like: $y(t) = \sin(2\pi \times 150 \times t)$ Why is there a $2\pi$ factor in the argument? When I ...
0
votes
1answer
444 views

Salary calculation: solving proportional increase of two variables

I would like to understand, proportionally, how much of someone's new salary is attributed to an increase in both their number of hours worked, and increased dolar wage respectively. Both these ...
1
vote
1answer
384 views

Number of solutions of Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
4
votes
3answers
780 views

The number of ones in a binary representation of an integer

Is there any relation that tells whether the number of ones in a binary representation of an integer is an even or an odd number?
2
votes
2answers
3k views

The equation of a line reflected about another line

I need to find the equation of this reflected line:
17
votes
3answers
603 views

How to answer a student objection to the use of “of” in pronouncing f(x)?

Once upon a time in elementary school, a student learned how to translate certain English words into math. For example, 'and' usually means 'plus' such as "If John has 3 oranges AND 5 apples, how ...
5
votes
2answers
143 views

Is $|x^r|=|x|^r$ for real numbers $x$ and $r$?

Suppose $x$ and $r$ are real numbers. Is $|x^r|=|x|^r$? If so, how do you prove it at the lowest level? (That is, using definitions and theorems available at K-12 level. If this is not sufficient ...
11
votes
5answers
410 views

How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$

Let $p \in \mathbf{N}$. I don't know how to prove that $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$ and $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$ ...
1
vote
1answer
78 views

Is the following substitution legitimate?

Suppose that $$ b_1 = const .f_1(x) \left( b_2 + \frac{a} {f_2(x)} \right) $$ becomes $$ b_1 = const. b_2 $$ because $f_1(x) \rightarrow 1$ and $f_2(x) \rightarrow \infty$ when $x \rightarrow ...
1
vote
1answer
73 views

Solving equations of the form $\operatorname{trig function}(x) = \mathrm{constant}$

Is there a way to solve for x where $$\operatorname{trig function}(x) = \mathrm{constant}$$ and where the domain is such that the function has an inverse. For example, $$\begin{align*}\sin x ...
0
votes
1answer
66 views

CORE 1, Expanding and Simplifying - HELP?

There's a question in my textbook: $$5x-6-(3x-2)$$ I think I misunderstood as I put the following for my working out: $$5x-6-3x-2$$ And from that I got: $$2x-8$$ Can anyone help me to understand ...
6
votes
1answer
256 views

given $a^2+b^2=28ab$ what's $\log_{3}\left (\frac{(a+b)^2}{ab}\right)$?

given $a^2+b^2=28ab$ what's $\log_{3} \left(\dfrac{(a+b)^2}{ab}\right)$? $\log_{3} \left(\dfrac{(a+b)^2}{ab}\right)$ $\log_{3} \left(\dfrac{a^2+b^2+2ab}{ab}\right)$ $\log_{3} ...
0
votes
1answer
144 views

Composition of functions

$f,g:\mathbb{R}\rightarrow\mathbb{R}$ $(f\circ g)(x)=3x-1$ a) $g(x)=2x-3; f(x)=?$ b) $g(f(x))=3x-6; g(\frac{1}{2})=?$
1
vote
1answer
54 views

Solving an equation for semiconductors

hey my lecturer put this example up for an exam tomorrow, could someone please explain how he gets to the 3rd line? is he using factorization? $$V_{\mathrm{dsb}}=V_{\mathrm{gsb}}-V_{\mathrm t}$$ ...
1
vote
2answers
141 views

Coins making up a certain sum

A cash register contains only dimes and quarters. There are $65$ coins equaling $\$12.80$ in the register. How many dimes and how many quarters are in the register?
3
votes
1answer
165 views

Inequality with absolute value

I am unsure if have solved the following inequality correctly: $ \dfrac{2x+3}{x+5} \leq \dfrac{x+1}{|x-1|} \tag{1}$ I've proceeded as follows. If $x>1$ then $|x-1|=(x-1)$ If $x<1$ then ...
2
votes
1answer
347 views

Cost of a can (check my answer please)

Question goes...A can is in the shape of a circular cylinder is required to have a volume of 750 cubic centimeters. The top and bottom are made of material that costs 8 cents per square centimeter ...
0
votes
1answer
82 views

what is $x^2+y^2+z^2=w^2$ when $x^2/y^2=y^2/z^2=z^2/x^2$?

What is the answer of $x^2+y^2+z^2=w^2$ when $x^2/y^2=y^2/z^2=z^2/x^2$ and $x, y , z, w\in\mathbb{N}$.
2
votes
2answers
115 views

$\log_{12} 2=m$ what's $\log_6 16$ in function of $m$?

Given $\log_{12} 2=m$ what's $\log_6 16$ in function of $m$? $\log_6 16 = \dfrac{\log_{12} 16}{\log_{12} 6}$ $\dfrac{\log_{12} 2^4}{\log_{12} 6}$ $\dfrac{4\log_{12} 2}{\log_{12} 6}$ ...
1
vote
1answer
46 views

Find M, since $\log_5 M = 2\log_5 A - \log_5 B+2$

Find M, since $\log_5 M = 2\log_5 A - \log_5 B+2$ I tried this: The answer is in function of A and B. $\frac{\log_M M}{\log_M 5} = 2\frac{\log_M A}{\log_M 5} - \frac{\log_M B+2}{\log_M 5}$ ...
2
votes
1answer
544 views

4 items add up to and multiply to 7.11 what are the value of the items?

A man walks into 711 and buys four items, the items add up to 7.11 and multiply to 7.11. What are the prices for the 4 items? During a talk about truth in mathematics, the presenter asked this ...
4
votes
3answers
286 views

Solving Radical Equations

This the Pre-Calculus Problem: $x-7= \sqrt{x-5}$ So far I did it like this and I'm not understanding If I did it wrong. $(x-7)^2=\sqrt{x-5}^2$ - The Square root would cancel, leaving: ...
1
vote
2answers
112 views

Algebra Sentence problem part b

Here is a new one I need help with. A dinner theater sells two types of tickets. Floor seats cost \$50 while stadium seats cost \$35. If the theater sold 170 seats for \$6625, how many tickets of ...
0
votes
1answer
51 views

Spheres and converting formulas

If the volume V of a sphere with radius r is V=(4/3)πr^3. If the surface area is s=4πr^2, how can I express the volume as a function of the surface area S? My first thought was to set them equal to ...
-1
votes
2answers
54 views

Algebra Sentence problem

I have a problem with this sentence. "If 9 is subtracted from the sum of two consecutive even integers, the result is 3 more than the larger of the integers. Find the integers."
0
votes
2answers
526 views

Solve simultaneous equations $2\log(y) = \log(2) + \log(x)$ and $2^y = 4^x$

I'm having trouble finding the solution to the following problem. (1) $2\log(y) = \log(2) + \log(x)$ (2) $2^y = 4^x$ So far all I've managed is: from (1) simplify using properties of ...
4
votes
2answers
505 views

How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?

This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...
1
vote
3answers
103 views

Why is it that $\left|b_n - b \right| < \frac{\left|b \right|}{2} \Rightarrow \left| b_n \right| > \frac{\left|b \right|}{2}$?

Unfortunately I am stuck on one step of a proof for an algebraic limit theorem, specifically: Why is it exactly that $\left|b_n - b \right| < \frac{\left|b \right|}{2} \Rightarrow \left| b_n ...
9
votes
4answers
426 views

Intuitive explanation of $(a^b)^c = a^{bc}$

What is an intuitive explanation for the rule that $(a^b)^c = a^{bc}$. I'm trying to wrap my head around it, but I can't really do it.
1
vote
1answer
862 views

Area of a square using circle

So I have this square and theres a circle inside of it. The circle of radius $r$ is inscribed in the square. So how do I find the area of the square in terms of $r$? I know that area of a circle is ...
4
votes
1answer
140 views

For all positive integers $n$, $14^{6n} - 11^{6n}$ is divisible by?

For all positive integers $n$, $14^{6n} - 11^{6n}$ is divisible by ? This question is followed with four options: $1)157\quad\quad 2) 163\quad\quad 3) 225\quad\quad \quad 4) \text{All ...
12
votes
1answer
262 views

If the product of $x$ positive integers is $n!$ What is the smallest possible value their sum can have?

If the product of $x$ positive integers is $n!$ What is the smallest possible value their sum can have? I was wondering what could be the most efficient strategy to solve this problem for ...
2
votes
1answer
280 views

Summation of natural number set with power of $m$

Who knows about the summation of this series: $$\sum\limits_{i=1}^{n}i^m $$ where $m$ is constant and $m\in \mathbb{N}$? thanks
7
votes
5answers
2k views

How to prove $\log n \leq \sqrt n$ over natural numbers?

It seems like $$\log n \leq \sqrt n \quad \forall n \in \mathbb{N} .$$ I've tried to prove this by induction where I use $$ \log p + \log q \leq \sqrt p \sqrt q $$ when $n=pq$, but this fails for ...
3
votes
3answers
121 views

Proving that for reals $a,b,c$, $(a + b + c)^2 \leq 3(a^2+b^2+c^2)$

Proving that for reals $a,b,c$, $(a+b+c)^2\leq 3(a^2+b^2+c^2)$. This is a homework question and I have no clue where to even start on this. I don't know if I am just tired or what but I can't get ...
8
votes
2answers
6k views

How to prove a limit exists using the $\epsilon$-$\delta$ definition of a limit

I understand how to find a limit. I understand the concept of the $\epsilon$-$\delta$ definition of a limit. Can you walk me through what we're doing in this worked example? It is from my student ...
1
vote
2answers
96 views

Typical age problem that I can't figure

Bob and Alice have together a sum of $103$ years old. In $4$ years, Alice will have $2$ times the age of Bob. What age will they have ? I'm trying different equation and doing my substitution but I ...
0
votes
2answers
1k views

The distance the Earth travels in one day

Find the distance that the Earth travels in one day in its path around the Sun. Assume that a year has 365 days and that the path of the Earth around the Sun is a circle of radius 93 million miles. ...
-2
votes
4answers
556 views

Solving the equation $\sin t = -\sqrt{2}/2$

Solving the equation $$ \sin t = -\frac{\sqrt{2} }{2} .$$ I know the solution is $1.25$ and $1.75$, but I do not know how to get there. An explanation would be GREATLY appreciated, thanks!