Algebra Word Problem Solutions

solve logarithmic equation without numerical methods
Is there algebraic method to solve following equation for x:
$ax+b\ln <!--\; \; -->x+c=0$
with $a,b,c$ constants without using numerical methods and ln means natural logarithm.

Graph the line that passes through (2,-1) parallel to the line 2x+3y=6

At the local grocery store, a 16-ounce bottle of apple juice costs $3.20. Find the cost of the apple juice per ounce

Show that $\mathrm{0.1248163264...}$ is irrational

Solve ${\displaystyle 2x+2=x^2-2}$

Modeling a center of mass of a thin wire
I am asked to find the moment about the x axis for a thin wire of constant density. This thin wire lies along the curve $y=\sqrt{x}$ and the limits for integration are $x=0$ and $x=2$.
I know from my textbook that the moment about the x axis is: $M_y=\int <!--\; \int \; -->\stackrel{~<!--\; ~\; -->}{y}dm$
Because this is a thin wire, I know that I need to subdivide the wire into small segments for integration. I have the following for relevant data for each segment:
Length: $dl=\sqrt{x}dx$
mass: $dm=\mathrm{\delta <!--\; \delta \; -->}dl=\mathrm{\delta <!--\; \delta \; -->}\sqrt{x}dx$
It's the part about the distance of the center of mass to the x axis that I think I'm missing. I have the following:
$\stackrel{~<!--\; ~\; -->}{y}=\sqrt{x}$
Therefore, my final integral is:
$M_y=\int <!--\; \int \; -->\stackrel{~<!--\; ~\; -->}{y}dm={\int <!--\; \int \; -->}_0^2\mathrm{\delta <!--\; \delta \; -->}\sqrt{x}\sqrt{x}dx=\mathrm{\delta <!--\; \delta \; -->}{\int <!--\; \int \; -->}_0^{2}xdx=\mathrm{\delta <!--\; \delta \; -->}{\frac{1}{2}{x}^2|}_0^2=\mathrm{\delta <!--\; \delta \; -->}2$
This particular problem is an odd numbered problem and so I know that I've got it incorrect. Please help me to see where I'm going wrong.

Finding the Maximum and Minimum of Trigonometric Equations By Quadratic Substitution
$y=3\cos <!--\; \; -->2x+7\sin <!--\; \; -->x+2$
The original question was: Finding the stationary point of the function. I tried to use both differentiation and trigonometric substitution of cos(2x) into $1-<!--\; -\; -->2{\sin }^2<!--\; \; -->(x)$.
For the quadratic method, I substituted sin(x) as u and solved for u, where the the equation is equated to 0, and finding the midpoint of the roots to find out the value of x following the nature of a parabolic equation.
Unfortunately I realized that trigonometric substitution eliminates a possible solution for x where $\cos <!--\; \; -->(x)=0$. I suspect that it might be due to a non function being converted into a function but I do not quite understand why does it causes so.

1.Determine the number of pounds of Julie’s body weight that are fat. Julie is your client that is 59kg and 22% body fat-Determine your lean body mass (FFM) in lbs-Subtract 1.0 from Julie’s desired percent body fat of 18%._____% (desired percent body fat) 1.0 - ._____ = ._____
-Determine your recommended body weight for Julie at that new % body fat.
2.How many pounds would she have to lose or gain to get to your recommended body weight?
2b.Do you think this is realistic, healthy goal? Why or why not?
3.What changes can Julie make in her diet and exercise routine to improve her body composition (to maintain or get to your recommended body weight)?

Equations that involve derivatives are called differential equations. Find at least one nonzero function that satisfies each of these differential equations. f'(x)=x

Find the next three terms of the arithmetic sequence 3,7,11,15,...

Edson exercises in sets that are each t minutes long. He does 6 sets of push-ups, 3 sets of pull-ups, and 4 sets of sit-ups. Write an expression to represent the time it takes Edson to exercise as the sum of three different terms. Then simplify the expression.

Lambert- W -Function calculation?
I have an equation of the form:
$n\log <!--\; \; -->n=x$
Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for it's evaluation.
Any ideas as to how I can evaluate?

Suppose $\gamma$ is a kth root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in <!--\; \in \; -->\mathbb{Z}$. Then $k=3,4$ or 6.
Let $k\in <!--\; \in \; -->\mathbb{Z}$ with $k>2$ and suppose $\mathrm{\gamma <!--\; \gamma \; -->}$ is a kth root of unity that satisfies a quadratic equation $z^2-<!--\; -\; -->mz-<!--\; -\; -->n=0$ with $m,n\in <!--\; \in \; -->\mathbb{Z}$. Then $k=3,4$ or 6.
My knowledge on algebra and number theory is poor, I just need this lemma to complete the computation of the analytic automorphism groups of complex tori.Can you help me prove this lemma with the knowledge of field theory and Euler phi function as small as possible? Actually I forget almost everything on field theory and have never learned number theory.

Correct notation for inequality equation. I'm currently taking a Polynomial functions -course and learning about quadratic inequality equations.
Let's consider this equation:
$-<!--\; -\; -->x^2+7x<10$
The correct answer to this according to our book is simply
$x<2$ or $x>5$
But is there any other way to express this in a more mathematical way? I understand why this is expressed the way it is as we haven't learned about set theory yet, but I'd like to know how it is expressed "correctly".

Is "$120\mathrm{\%<!--\; \%\; -->}$" an integer or irrational number ?

The grade appeal process at a university requires that a jury be structured by selecting six individuals randomly from a pool of nine students and eight faculty.​ (b) What is the probability of selecting a jury of all​ faculty?

When can I use the natural log to help solve an integral?
Why is it okay to do this: $\int <!--\; \int \; -->\frac{1}{x-<!--\; -\; -->2}dx=\ln <!--\; \; -->(x-<!--\; -\; -->2)$
but not this: $\int <!--\; \int \; -->\frac{1}{1-<!--\; -\; -->x^2}dx=\ln <!--\; \; -->(1-<!--\; -\; -->x^2)$

Find the sum of the geometric sequence -1, 6, -36, ... if there are 7 terms

How do you find ${\displaystyle a_{25}}$ given ${\displaystyle a_n={(-1)}^n(3n-2)}$?

For what value of k are the graphs of 12y=−3x+8 and 6y=kr−5 parallel? For what value of k are they perpendicular?