Algebra Word Problem Solutions

Find the solution of the differential equation that satisfies the given initial condition?
$y(x+1)+y^{\prime }=0$, $y(-<!--\; -\; -->2)=1$
I don't understand what this question is telling me to do? Can someone please explain?

Which shows the graph of the solution set of 3y – 2x > –18?

Find k so that the following function is constant on any interval
$f(x)=\{\begin{array}{ll}{x}^2+k& \text{if }x\le <!--\; \le \; -->5\\ kx& \text{if }x>5\end{array}$

Which shows the following expression after the negative exponents have been eliminated?
$\frac{x{y}^{-<!--\; -\; -->6}}{x^{-<!--\; -\; -->4}{y}^2}\Rightarrow <!--\; \Rightarrow \; -->x^{1+4}{y}^{-<!--\; -\; -->6-<!--\; -\; -->2}$

What is the factored form of $x^2-4x-5$?
$$\begin{gathered} (x+5)(x-1) \\ (x+5)(x+1) \\ (x-5)(x-1) \\ (x-5)(x+1) \end{gathered}$$

We have the equation y - 3x = -2
Select true or false
The system of equations has no solution
The system of equations has only one solution
The system of equations has infinitely many solutions
The point (-1,-5) is a solution to the system of equations
The two lines in the system of equations have the same slope

How can the expression $(x)(3x{)}^3$ be simplified?

How to factor $6x^2-<!--\; -\; -->5x+1$

Solve for x.
$40=\frac{x}{-<!--\; -\; -->\frac{4}{7}}$

We need to determine if the graph is symmetric about the $x-$axis, the $y-$axis, or the origin.$r=2\cos 5\theta$?

Alternate proof for "${\log }_{10}<!--\; \; -->2$ is irrational"
I need to prove that ${\log }_{10}<!--\; \; -->2$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and wanted to check its validity!
Prove by contradiction: Suppose that $\log <!--\; \; -->2$ is rational - that is, it can be written as
$\log <!--\; \; -->2=\frac{a}{b}$
where a and b are integers. Then
$2={10}^{\frac{a}{b}}$
$2={10}^a{10}^{\frac{1}{b}}$
$\frac{2}{{10}^a}={10}^{\frac{1}{b}}$
Log both sides:
$\log <!--\; \; -->(\frac{2}{{10}^a})=\frac{1}{b}$
$\log <!--\; \; -->2-<!--\; -\; -->\log <!--\; \; -->({10}^a)=\frac{1}{b}$
$\log <!--\; \; -->2=\frac{1}{b}+a$
$\log <!--\; \; -->2=\frac{ab+1}{b}$
However we assumed that $\log <!--\; \; -->(2)=\frac{a}{b}$ and thus we have a contradiction.

Every integer is a rational number.
True or false

More precise way of solving inequality
I need to solve this function:
$|<!--\; |\; -->x^2-<!--\; -\; -->1|<!--\; |\; -->\ge <!--\; \ge \; -->2x-<!--\; -\; -->2$
I solved this equation:
For $x<0$, the solution is non existing, here I got negative root, when I tried to solve quadratic function and for $x\ge <!--\; \ge \; -->0$ I got points $x_1=-<!--\; -\; -->1$ and $x_2=3$.
My question is:
How do I set the solution of equation. Is there any procedure, with wich I can determine is equation valid for $[-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},-<!--\; -\; -->1]\vee <!--\; \vee \; -->[3,+\mathrm{\infty <!--\; \infty \; -->}]$.
I know that I can just set the numbers and see the result, but I just want to now is there any other different way to do this.

Modeling phenomena using random variables
A load balancer in a cloud computing system is composed of N servers. The balancer, When assigning the connections to the servers, decide at random which one to use. But nevertheless, Due to some prioritization of the servers, 10% of them have a higher utilization level than the rest. Model the problem of access to the servers of the cloud computing system, for that purpose define clearly the sample space ($\mathrm{\Omega <!--\; \Omega \; -->}$), the observations in the sample space ($\mathrm{\omega <!--\; \omega \; -->}$) and the probability function of the events of the problem. Also define an impossible event. Finally, define the random variables X and Y, which respectively represent the events of accessing a system server and accessing any server of the most used. Which of these random variables gives more information about the problem?

Infinite series representation for root of polynomials?
Given a polynomial $p(x)=a_n{x}^n+\cdots <!--\; \cdots \; -->+a_{1}x+a_0$, can every root of the polynomial be represented as $\underset{k=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{b}_k$ with the $b_k$'s being a function of $a_0,\dots <!--\; \dots \; -->,a_n$ using only elementary operations of arithmetic and taking roots?

Write a rule for the nth term of the arithmetic sequence given ${\displaystyle a_{20}=240,a_{15}=170}$

Find all the ages when the son will be as old as his father today, the sum of their ages then will be 126 and when the father was as old as his son is today, the sum of their ages was 38

Find the common ratio in this geometric sequence 3, 18, 108, 648, 3888, ...

Prove the inequality $\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{c}{1+c^2}\le <!--\; \le \; -->\frac{3\sqrt{3}}{4}$
$a,b,c$ are nonnegative real numbers such that $a^2+b^2+c^2=1$
Prove the inequality
$\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{c}{1+c^2}\le <!--\; \le \; -->\frac{3\sqrt{3}}{4}$
I tried the method of Lagrange multipliers and Jensen's inequality but I have not been proved this inequality

The measure of a supplement of an angle is 12° greater than three times the measure of a complement. How do you find the measure of the angle?