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.
47,624
questions
-4
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1
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can we prove $0 < q < 1$ implies $0 < 1- q < 1$ algebraically WITHOUT multiplying by $-1$? [closed]
can we prove $0 < q < 1$ implies $0 < 1- q < 1$ WITHOUT multiplying by $-1$?
Seems like we can't do it. I know it's true, and it's easy to put together word arguments like "the ...
2
votes
1
answer
82
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If $M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$ How to Find $\frac{n^3+2n}{M}$?
$$M:=\sum\limits_{k=1}^{\frac{n(n+1)}{2}}\lfloor\sqrt{2k}\rfloor$$
Find $\frac{n^3+2n}{M}$
This problem was on a problem book.
It is easy to find $M$
If $n$ is odd, $\ m=\frac{n+1}{2} $ and $$M= \...
- 6,620
0
votes
1
answer
57
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Show that $\left| x-y \right| \leq \left| x \right| + \left| y \right|$ for all real numbers $x$ and $y$ [duplicate]
Show that $\left| x-y \right| \leq \left| x \right| + \left| y \right|$ for all real numbers $x$ and $y$
By definition, $-|x| \leq x \leq |x|$ and $-|y|\leq y \leq |y|$.
$\Rightarrow -|x|+|y| \leq x-...
4
votes
7
answers
108
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Solve $\left| 2x-5 \right| \leq \left|x+4 \right|$
Solve $\left| 2x-5 \right| \leq \left|x+4 \right|$
If both are positive: $2x-5 \leq x+4 \Rightarrow x \leq 9$
If one is negative: $2x-5 \leq -(x+4) \Rightarrow 2x-5 \leq -x-4 \Rightarrow x \leq \frac{...
1
vote
3
answers
78
views
Solve $\left| 3x+2\right| \geq 4$
Solve $\left| 3x+2\right| \geq 4$
If $3x+2 \geq 0$:
$3x+2 \geq 4 \Rightarrow x \geq \dfrac{2}{3}$
If $3x+2 < 0$:
$-(3x+2) \geq 4 \Rightarrow -3x-2 \geq 4 \Rightarrow -x \geq 2 \Rightarrow x \leq -...
2
votes
0
answers
30
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Is a function whose graph is central symmetric with respect to any point always linear? [duplicate]
If $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(x+h)-f(x)=f(x)-f(x-h)$ for every $x \in \mathbb{R}, h \geq 0$, must $f$ be of the form $f(x)=ax+b$?
I originally thought to define $f(x)=\frac{1}{...
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-2
votes
4
answers
74
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I need an equation that will net me 97% of a whole after a 2.9% + 0.30 cent fee [closed]
Say I want to charge my customer 100.00. I need to add a fee on top of the 100.00 so that after a 2.9% credit card fee + 0.30c flat fee is taken from the 100.00, I end up with 97.00 (97%) Net
What ...
- 111
2
votes
1
answer
105
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An alternative solution to the equation $\sqrt{\sqrt{x+2}+\sqrt{x-2}}=2\sqrt{\sqrt{x+2}-\sqrt{x-2}}+\sqrt{2}$
Solve the equation
$$\sqrt{\sqrt{x+2}+\sqrt{x-2}}=2\sqrt{\sqrt{x+2}-\sqrt{x-2}}+\sqrt{2}$$
I know how to solve this equation using double substitution $\sqrt{x+2}=a$, $\sqrt{x-2}=b$. But I want to ...
- 1,433
4
votes
5
answers
156
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Solve $\left| \dfrac{2x-1}{x+1 }\right|=3$
Solve $\left| \dfrac{2x-1}{x+1 }\right|=3$
$\Rightarrow \dfrac{\left| 2x-1 \right|}{\left| x+1 \right|}=3$
$\Rightarrow \left| 2x-1 \right|=3\left| x+1 \right|$
If both $\left| 2x-1 \right|$ and $\...
1
vote
1
answer
43
views
Find the value of y from the expression $0 =\dfrac{1}{y^2} - x^2 + 2Cx^3 - \dfrac{4aC}{y^3}x$. [Ans. $ y = \dfrac{1}{x} + C - 2aCx $]
Here is my attempt.
$$\begin{align*}
& 0 =\dfrac{1}{y^2} - x^2 + 2Cx^3 - \dfrac{4aC}{y^3}x\\
\Rightarrow & \dfrac{1}{y^2} = x^2 - 2Cx^3 + \dfrac{4aC}{y^3}x \\
\Rightarrow & \dfrac{1}{y^2}...
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-1
votes
0
answers
45
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How do we create this linear equation?
Given two pair of equations:
Pair 1
$(.5+.5r)(.5-r)=A_0$
$(.5+.5r)^2(.5-r)=B_0$
Pair 2
$(r-.5)(1-.5r)=A_0$
$(r-.5)^2(1-.5r)=B_0$
We are given a pair of two values $A_0, B_0$ that satisfy Pair 1 or ...
0
votes
0
answers
22
views
Limitation of functions in describing sequences in condensed form. [closed]
Are there boundaries till which a function or a combination of functions can explain another function? If so which branch of mathematics deals with it?
For example- The Fibonacci sequence has a ...
4
votes
2
answers
97
views
ACT practice test, aren't both $3$ and $12$ viable answers? [closed]
The question
For which of the following values of $c$ will there be two distinct real solutions to the equation $5x^2+16x+c=0$?
and the possible answers are:$\quad$
$\text{F}.\space3\\
\text{G}.\...
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0
votes
1
answer
103
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Where does the third solution come from?
There's a well known trick with polynomials. If we have
\begin{align*}
&&(x-r_1)(x-r_2) &= 0 \\
&& x^2 -(r_1+r_2)x + r_1r_2 &=0 \tag{*}\label{*} \\
&\text{so}& x^2 &...
- 1,509
0
votes
0
answers
47
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I need help finding the upper and lower bounds of a polynomial's roots [closed]
I used rational root theorem to obtain the possible roots of
$f(x)=-x^4+3x^3-4x^2-7x+9$
The upper boundary I obtained from rational root theorem was $3$. When I used $3$ in synthetic division I got ...
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