Improve Your Algebra Skills with Practice Problems

Given expression is,
${\displaystyle \frac{15}{21}=\frac{130.2}{93}}$
We need to determine whether the ratios are equal.

In 2000, ${\displaystyle 51\mathrm{\%}}$ of the residents in a large city regularly used newspapers for getting news and this has decreased at an average rate of approximately ${\displaystyle 1.9\mathrm{\%}}$ per year since then. Find a linear function in slope-intercept form that models this description. The function should model the percentage of residents, P(x), who regularly used the news outlet xyears after 2000.
${\displaystyle P\left(x\right)=-1.9x+51}$

(a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state the value of and ${\displaystyle \delta \left(\text{or}M\text{or}N\text{and}\delta \right)}$ in each case.
(i) How close to 4 do we need to take x so that ${\displaystyle (\frac{x}{2}-2)<0.001}$ ?
(ii) How close to 4 do we need to take x so that ${\displaystyle (\frac{x}{2}-2)>0.0001}$ ?
(iii) How close to 0 do we need to take x so that ${\displaystyle (2x+9)<9.0001}$ ?
(iv) How close to 0 do we need to take x so that ${\displaystyle (2x+9)>8.999}$?
(v) How close to 0 do we need to take x so that ${\displaystyle (x^2+6x+9)<9.001}$ ?
(vi) How close to 0 do we need to take x so that ${\displaystyle (x^2+6x+9)<9.0001}$ ?
(vii) How close to 0 do we need to take x so that ${\displaystyle (x^2+6x+9)>8.9999}$ ?
(viii) How close to -7 do we need to take x so that ${\displaystyle \frac{1}{{(x+7)}^4}>10000}$ ?
(ix) How close to −7 do we need to take x so that ${\displaystyle \frac{1}{{(x+7)}^4}>100000}$ ?
(x) How close to 0 do we need to take x so that ${\displaystyle \ln x<-10000}$?
(b) Use the definition of a limit to show that
(i) ${\displaystyle limx\to 4}$
${\displaystyle (\frac{x}{2}-2)=0}$
(ii) ${\displaystyle limx\to 0}$
${\displaystyle (2x+9)=9}$
(iii) ${\displaystyle limx\to 0}$
${\displaystyle (x^2+6x+9)=9}$
(iv) ${\displaystyle limx\to -7}$
${\displaystyle \frac{1}{{(x+7)}^4}=\mathrm{\infty }}$
(v) ${\displaystyle limx\to 0+}$
${\displaystyle \ln x=-\mathrm{\infty }}$

A bank finds that the estimated proportion of clients defaulting on a loan, given the interest rate is below ${\displaystyle 15\mathrm{\%}}$, is 0.34, They also find that the estimated proportion of clients defaulting on a loan, given the interest is greater than or equal to ${\displaystyle 15\mathrm{\%}}$, is 0.52.
a) What is the odds ratio of defaulting given the interest rate is greater than or equal to ${\displaystyle 15\mathrm{\%}}$ relative to the interest rate is lower than ${\displaystyle 15\mathrm{\%}}$? Interpret this odds ratio.
b) If we were to analyze this data using the following logistic regression model, what are the estimates of ${\displaystyle {\beta }_0}$ and ${\displaystyle {\beta }_1}$? Show your work.
${\displaystyle lg\left\{\left(\frac{p}{1-p}\right)\right\}={\beta }_0+{\beta }_{1}x}$
Where p is the probability of defaulting on a loan and x is an indicator variable that is 1 when the interest rate is greater than or equal to ${\displaystyle 15\mathrm{\%}}$ and 0 when the interest rate is less than ${\displaystyle 15\mathrm{\%}}$.

Calculate the unit rate for per pound if a 15 Ib. bag of potatoes cost $3.99.

Solve the exponential equation. Express irrational solutions as decimals correct to the nearest thousandth.
${\displaystyle 4^{x-1}=3^{2x}}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is ______.
B. The solution is the empty set.

The cost in dollars to produce x t-shirts is ${\displaystyle C\left(x\right)=3.56x+80}$. The revenue in dollars from sales of x t-shirts is ${\displaystyle R\left(x\right)=9.34x}$.
a) Write and simplify a function P that gives profit in terms of x.
b) Find the profit if 100 t-shirts are produced and sold.
a) ${\displaystyle P\left(x\right)=?}$
b) The profit if 100 t-shirts are produced and sold is $?

Factor
1) $-36m+16$
2) $-21a+35$
3) $-30n-10$
4) $30x+70$

Using data from 2009 through 2015, the percent of households with internet access in a certain region can be modeled by the function ${\displaystyle f\left(x\right)=51.952x^{0.16}}$ to the number of years after 2005.
a. Find the formula for the inverse of the function f.
b. If the inverse is ${\displaystyle f^{-1}}$, find ${\displaystyle f^{-1}.\left(f\left(20\right)\right)}$

Simplify the expression:
${\displaystyle \frac{1}{-3+i}}$

Write the following ratio proportions as fraction proportions.
${\displaystyle 75÷100=150÷200}$

1. Convert 2.5 pt. to milliliters by Ratio and Proportion.
2. Convert 6.2 gr to milligrams by Ratio and Proportion.

Determining the value of x in a proportion:
Write the proportion with the known digits on the LHS (left hand side) of the equation, whereas the unknown quantity should be written on the RHS (right hand side) of the equation.
For example,
${\displaystyle \frac{5}{15}=\frac{x}{21}}$

Consider that the number of student ticket sold for a game is ${\displaystyle S\left(p\right)=59p+8100}$ and number of non student ticket sold is given by ${\displaystyle N\left(p\right)=0.2p^2+12p+4100}$, here p denotes the winning percentage.
a) Equation of total number of tickets sold T(p) for the game.
b) Find the domain of the function ${\displaystyle T\left(p\right)=0.2p^2+71p+12200}$
c) Given for ${\displaystyle p=90\mathrm{\%}}$ stadium is full. Find the capacity of stadium.

Define what is meant by the term ratio. Compare and contrast rates, proportions, and percentages. Give an example of each one

David drives to school in rush hour traffic and averages 32 mph. He returns home in mid-afternoon when there is less traffic and averages 48 mph. What is the distance between his home and school if the total traveling time is 1 hr 15 min?

Rationalize the expression:
${\displaystyle {\left(4xy\right)}^{-1}}$

There's a roll with 40 coins. A coin officially weighs 5g. The actual weight may be 0.194g off. The roll itself weighs 1.5g. What is the range of possible wights for the roll?

Solve the proportion for the variable by cross-multiplying.
Given: ${\displaystyle \frac{-12}{13}=\frac{36}{x}}$

To find that how long should be the model if the carbose is 30 feet long.