Algebra Word Problem Solutions

Are certain measurements always irrational numbers?

How many numbers less than $x$ have a prime factor that is not $2$ or $3$
I am trying to figure out the number of integers greater than 1 and less than or equal to x that have a prime factor other than 2 or 3. For example, there are only two such integer less than or equal to 7.
It is straight forward to determine how many many integers less than or equal to x have a prime factor other than 2:
$x-<!--\; -\; -->\lfloor {\log }_{2}x\rfloor$
Or to make the same determination about 3:
$x-<!--\; -\; -->\lfloor {\log }_{3}x\rfloor$
What is the method or formula for figuring out how many integers less than or equal to x have a prime factor other than 2 or 3?
I know that it is less than:
$x-<!--\; -\; -->\lfloor {\log }_{2}x\rfloor -<!--\; -\; -->\lfloor {\log }_{3}x\rfloor$
and greater than:
$x-<!--\; -\; -->\lfloor {\log }_{2}x\rfloor -<!--\; -\; -->\lfloor {\log }_{3}x\rfloor -<!--\; -\; -->\lfloor \frac{x}{6}\rfloor$
Thanks

A bag of grapes is being divided into 24 smaller bags of 4 ounces each for snacks during a field trip. What is the division equation to find the number of ounces in the larger bag?

Why is a machine producing 7 nails in 9 seconds faster than one producing 11 nails in 14 seconds?
Machine A Produces 11 nails in 14 seconds.
Machine B produces 7 nails in 9 seconds.
Which is faster?
I thought that $\frac{11}{14}$ is larger than $\frac{7}{9}$ and thus A is the answer. However, the answer says that $\frac{9}{7}$ is bigger than $\frac{14}{11}$ so B is the answer. How does this work?

Using logarithms to solve the following equation to find x
$9^{2x}={27}^{1-<!--\; -\; -->x}$ ?? I'm really struggling with this questions. I appreciate your help and if you can please show me your working out so I can understand it too

Find the slope of line parallel to 2y=6x+8

Find the fifth term of a geometric sequence on a calculator

Solve logarithmic equation for $x$ to find the inverse of $f(x)=\ln <!--\; \; -->(x+\sqrt{x^2+1})$

If ${\displaystyle a_0=7}$ and ${\displaystyle a_{n+1}=a_n+12}$ for ${\displaystyle n\ge 0}$, how do you find the values of ${\displaystyle a_5}$?

Kay spends 250 min/wk exercising. Her ratio of time spent on aerobics to time spent on weight training is 3 to 2. How many minutes per week does she spend on aerobics? How many minutes per week does she spend on weight training?

Find the solution to the differential equation
Assume $x><!--\; >\; -->0$ and let
$x(x+1)\frac{du}{dx}=u^2,$
$u(1)=4.$
I started off by doing some algebra to get:
$\frac{1}{u^2}du=\frac{1}{x^2+x}dx.$
I then took the partial fraction of the right side of the equation:
$\frac{1}{u^2}du=(\frac{1}{x}-<!--\; -\; -->\frac{1}{x+1}).$
I then took the integral of both sides:
$-<!--\; -\; -->\frac{1}{u}=\log <!--\; \; -->x-<!--\; -\; -->\log <!--\; \; -->(x+1)+C.$
From here I don't know what to do because we are solving for $u(x)$ and I'm not sure how to get that from $-<!--\; -\; -->\frac{1}{u}$

I need two numbers that multiply to get -20 and add to get 1. I'm doing a quadratic functions problem and I can't find the answer to this part of the question.
It started out as finding the zero of the equation by factoring. The equation was $g(x)=2x^2+x-<!--\; -\; -->10$. I set it equal to zero, $x^2+x-<!--\; -\; -->20=0$. Now I have to factor but I can't find a way to. If it's prime, I don't know how to answer the main question.
$a\times <!--\; \times \; -->b=-<!--\; -\; -->20$ and $a+b=1$

How to find explicit formula for a sequence seems to be geometric but isn't?
I recently came across this interesting problem, where the progression was neither arithmetic nor geometric. The problem asks for an explicit formula for the following recursive formula:
$a_1=2$
$a_n=5(a_{n-<!--\; -\; -->1}+2)$
So the first five terms are 2,20,110,560,2810.
I tried to distribute the 5 in the second equation to get $5a_{n-<!--\; -\; -->1}+10$ and then tried to apply the formula for a geometric sequence, which is $a_1(r{)}^{n-<!--\; -\; -->1}$, but got only $2(5{)}^{n-<!--\; -\; -->1}$, which clearly doesn't work for the sequence.
I also tried using finite differences on the first five terms to see if it was just a polynomial rule, but that didn't work.
How would you go about doing a problem like this?

Dealing with Logarithms. $\log <!--\; \; -->(b^x+a)=\log <!--\; \; -->(c)$
What methods/techniques are available to solve for x in the following type of situation:
$\log <!--\; \; -->(b^x+a)=\log <!--\; \; -->(c)$
The only log methods I have been exposed to are using the power laws and bring x out, which you cannot do in this case.
Thanks for your help.

Modeling the relationship between perimeter and area
Is there any equation that models the relationship between the area and perimeter of a rectangle?

I need to find the sum of the arithmetic sequence 3, 9, 15…, if there are 24 terms

What kind of a triangle is it if the angles of a triangle are represented by x, 3x + 20, and 6x

If now we have two irrational numbers $\mathrm{\alpha <!--\; \alpha \; -->}$ and $\mathrm{\beta <!--\; \beta \; -->}$, can we find sequences $\{a_n\}$, $\{b_n\}$ and $\{q_n\}$ with $q_n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$ as $n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$ such that
$|\mathrm{\alpha <!--\; \alpha \; -->}-<!--\; -\; -->\frac{a_n}{q_n}|<\frac{1}{q_n^2}\text{ and }|\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->\frac{b_n}{q_n}|<\frac{1}{q_n^2}?$

Find the slope of a line parallel to 10x−5y=8

Show that $|\mathbb{I}|=|\mathbb{R}|$