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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 topics.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Domain of $\sqrt{1+\frac1x}$

I solved this as follows: $1+\frac1x \ge 0$ $\frac1x \ge -1$ (subtract $1$ from both sides) $1 \ge -x$ (multiply $x$ to both sides, cancel out $x$ from bottom of left side) $-1 \le x$ (multiply both sides by $-1$, change sign) $x \ge -1$ Another way…
user5826
  • 11,982
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Can I write $(x^p)^{q} = x^{pq}$, where either of $p$ or $q$ are rational.

I am confused with a very basic algebra question about the following law of exponents. We know that $(x^n)^{m} = x^{nm}$, holds true for real $x$ and integer exponents $n, m$. I want to know whether this result holds for rational exponents…
mathscrazy
  • 1,837
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Meaning of Problem in Evaluation

In Algebra, a good rule-of-thumb I saw was If solving an equation leads to a contradiction, there is no solution. And this makes sense to me, particularly in the following case: $x-1 = x +1 \Rightarrow -1 = 1$ We are being asked to find a number…
user70530
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Next step in solving an equations...

I have managed so far to break down an the following equation: $x^n+y^n=1$ to $x^n=1-y^n$ but what is the next step to get $x$ on it's own? I have hopped over here from StackOverflow where I am trying draw superellipse where a and b are always 1. So…
Ross
  • 143
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why substitution works in general

Consider the equations $x^2 +y^2=1$ and $y=x^2-1$. If $y=x^2-1$ is substituted into $x^2 +y^2=1$ to obtain $x^2 +(x^2-1)^2=1$, then the $x$ values that result from solving this equation will be the $x$ values that produce equal $y$ values in the…
user796511
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How is the last equation derived in this equality?

From Inequalities A Mathematical Olympiad Approach, page 122 Observe that, $$\sum_{i = 1}^n \sqrt{a_i}-(n-1)\sum_{i = 1}^n \dfrac{1}{\sqrt{a_i}} = \sum_{i = 1}^n \dfrac{1}{1+a_i}\sum \sqrt{a_i} - \sum_{i = 1}^n \dfrac{a_i}{1+a_i} \sum_{i = 1}^n…
EHN
  • 91
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De Morgan's saying: "I was x years old in the year x^2"

Augustus de Morgan, when asked about his age, remarked: I was $x$ years old in the year $x^2$. He died in $1871$. Examining possible squares, we find $41^2 = 1681$, $42^2 = 1764$, $43^2 = 1849$, $44^2 = 1936$. Clearly, the only one which makes…
Dimitris
  • 767
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Looking for a function which is convex when x is between 0 and 1 but concave when x is greater than 1

I am looking for such a function for plotting a figure like in the picture in Python. Many thanks!
Linghui
  • 41
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How many employees do I need to fill 33 positions? Real world problem

thanks for reading this thread! This is a problem which I don't know how to put in a formula properly: We're filling 33 positions with workers across a full year (12 months). Each worker will take 1 month off during the year. We will need to fill…
Adhurim Uka
  • 43
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Elementary doubt on ratios

If $y=-x$ and $\displaystyle \frac{y}{x-z}=\frac{x}{y}$ then either $x:y:z=1:-1:0$ or $x:y:z=-1:+1:0$. Is this correct? If not why?
Quixotic
  • 22,431
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Find $S$ where $S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$, why am I getting an imaginary number?

$\large S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$ Multiplying by conjugate: $\large S=\dfrac {-3}{\sqrt[3] {5+2 \sqrt {13}}-\sqrt[3]{5-2 \sqrt {13}}}$ From the original: $\large S-2\sqrt[3]{5-2 \sqrt {13}} =\sqrt[3] {5+2 \sqrt…
Ovi
  • 23,737
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Is it possible to evaluate $232^2-62^2\times14$ by factoring or using identities or...?

The expression $232^2-62^2\times14$ can be calculated directly ($53824-3844 \times14=8$). But is it possible to evaluate it for example by factoring or using identities? Here is what I have…
Etemon
  • 6,437
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How many more men than women were there in the beginning?

Let men=m, women=w, children=c From the statement, m=1.25w 0.8w=c Children left resulted in equal number of men and women, women=122 at the end. Can I assumed that the total number of people is…
LTY
  • 123
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How to solve $\\e^{-u}+\frac{u}{5}=1\\$ for $\\u\\$ ? What is the method to solve it without using graph.

Solving $\\e^{-u}+\frac{u}{5}=1\\$ without using graph. From graphing line $y(u)=1$ intersects with $\\e^{-u}+\frac{u}{5}\\$ at two points (0,1) & (4.97,1) so that gives $\\u = 0\\$ or $\\4.97\\$. But how can I solve it analytically ? [[graph]
Salz Engineering
  • 109
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How to find solutions to this equality $\; \mathrm{x} = \mathrm{a^2x \, (1-x)\,(1-ax\,(1-x))}$

We have the following equality: $$ \mathrm{x} = \mathrm{a^2x \, (1-x)\,(1-ax\,(1-x))}$$ Some of the solutions I found: $\mathrm{x} = 0$ Also for $\mathrm{a}=0$, every $\mathrm{x}$ is a solution I believe I tried getting everything out of the…
Prael
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