Improve Your Algebra Skills with Practice Problems

Explain in your own words what it means for an equation to model a real-world situation, and give an example.

Identify the expression as an equation containing fractions, a fractional equation, or a proportion.
${\displaystyle \frac{x}{12}=\frac{17}{3}-\frac{2x}{9}-5+\frac{3x}{12}}$
1) Fractional equation
2) Equation containing fractions
3) Proportion

Solve the equation ${\displaystyle 15x^2+2x-8=0}$.
Answer: ${\displaystyle z=\left(\frac{2}{3}\right)(-\frac{4}{5})}$
Write your answer as a list of integer or reduced fractions, with your answers separated by (a) comma(s).
For example, if you get 4 and ${\displaystyle -\frac{2}{3}}$ as your answers, then enter ${\displaystyle 4,\frac{2}{3}}$ in the bow.

Add the fractions. If possible, simplify your answer.
${\displaystyle \frac{7}{a^2+6a+9}+\frac{8}{a^2-9}=?}$

Please explain the addition property of equality with fractions

Solve for the indicated variable. Fully reduce fractions.
a.) Solve ${\displaystyle 2L+2W=P2L+2W=P}$ for W
b.) ${\displaystyle 2xyz=242xyz=24}$ for y

Solve and check the following equation.
${\displaystyle \frac{x-7}{5}+\frac{1}{5}=-\frac{x}{10}}$
Select the correct choice below and fill in any answer boxes in your choice.
A. The solution set is (_).
B. The solution is the empty set.

Simplify the following expressions using the imaginary number i:
1. ${\displaystyle 5\sqrt{-12}}$
2. ${\displaystyle \sqrt{-8}}$
3. ${\displaystyle 3\sqrt{-7}}$
4. ${\displaystyle -3\sqrt{-200}}$
Write the following numbers using the imaginary number i, and then perform the operations necessary and simplify your answer.
1. ${\displaystyle 3\sqrt{-9}\cdot 2\sqrt{-49}}$
2. ${\displaystyle \sqrt{-12}\cdot \sqrt{-36}}$
3. ${\displaystyle i\sqrt{-36}-i2}$
Simplify the following expressions using the imaginary number i when needed:
8. i8
9. i75
10. (i2)(i6)
Simplify and rationalize the denominators.
11. ${\displaystyle 4\sqrt{-\frac{3}{2}}}$
12. ${\displaystyle -3\sqrt{-\frac{2}{27}}}$
13. ${\displaystyle \sqrt{-\frac{4}{27}}}$
Perform the operation indicated in each problem below:
14. ${\displaystyle (8+6i)+(2+i)}$
15. ${\displaystyle -9+(4-2i)}$
16. ${\displaystyle (1+i)-(1-i)}$
17. ${\displaystyle 3i-(4+15i)}$
Perform the operation indicated in each problem below:
18. ${\displaystyle (\sqrt{2}+3i)(\sqrt{2}-3i)}$
19. ${\displaystyle (\sqrt{27}-3i)(\sqrt{3}+i)}$
20. ${\displaystyle (\sqrt{2}-i\sqrt{3})(\sqrt{2}+i\sqrt{3})}$
21. ${\displaystyle 2i(3i-2)}$
22. ${\displaystyle (3-6i)2}$
Determine the sum or difference in geometric terms, and then check it algebraically. Remember that ${\displaystyle a-b=a+\left(b\right)}$.
23. ${\displaystyle (8-5i)-(2+3i)}$
24. ${\displaystyle (-6+i)-(-2-3i)}$
25. ${\displaystyle (3+i)+(-2+2i)+(8-2i)}$
Solve the following equations using the quadratic formula.
26. ${\displaystyle X2-4X+13=0}$
27. ${\displaystyle X2+49=0}$
28. ${\displaystyle X2+5X+8=0}$
29. ${\displaystyle X2-3X+4=0}$

What is wrong with the following equation: ${\displaystyle 4+\left(\frac{10}{5}\right)=6-2}$
A) Equations should not involve expressions on both sides.
B)The equation mixes different operations like +, / and -.
C) The left sides

Solve a ratio problem in various ways, such as make a table, draw a diagram, use a unit rate, solve an equation, and write a proportion. The aspect ratio describes the shape of a picture. It is a ratio of the width (horizontal direction) to the height (vertical direction). A photographer has a picture with an aspect ratio of 3 to 5. Can the photographer fit the picture exactly on paper that has a width of 12 inches and a height of 20 inches? Solve this problem using the Unit Rate strategy.

Find the domain of the function.
${\displaystyle f\left(x\right)=\sqrt{x+4}-\frac{\sqrt{1-x}}{x}}$

Statement showing a relationship between numbers that are not necessarily equal using the symbols ${\displaystyle >,<,\le ,\ge ,\text{or}\ne q}$ are called _.
1) inferences
2) inconsistencies
3) unequals
4) inequalities

Consider the following statement: Mary wants to (spend, at least spend, at most spend) 20 minutes a day exercising. Depending on which word you chose in parenthesis you end up with a different expression.
Write the expression you would get for each of the word choices and label them.

Adding Decimals with Different signs. In January, the weight of a lion at a zoo changed by 1.8 pounds. In February, its weight changed by -2.3 pounds. What was the overall increase or decrease in the lion's weight?

Emily needs enough fabric for ${\displaystyle 3\frac{1}{2}}$ hats. It each hat requires ${\displaystyle 1\frac{3}{5}}$ feet of fabric, how much fabric will she need to make the ${\displaystyle 3\frac{1}{2}}$ hats?

Let ${\displaystyle y=f\left(x\right)}$ be a function $D=(-12,13)$ and range ${\displaystyle R=(-\mathrm{\infty },\mathrm{\infty }).}$ Find the domain D and range R for the following functions and enter your answers using interval notation. Keep in mind order of operations. Be sure your intervals are in the correct order, and enter exact answers only (no approximations).
a) ${\displaystyle y=f\left(\frac{1}{2}x\right)}$
b) ${\displaystyle y=5f\left(2x\right)-2}$

Solve the exponential equation. Express the solution set so that (a) solutions are in exact from and, if irrational, (b) Support your solutions by using a calculator.
${\displaystyle 3{\left(3\right)}^{x-3}+2=77}$
a) 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.
b) 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.

Simplify the following expression:
${\displaystyle -5+i+9-6i}$

You should answer this question WITHOUT using a CALCULATOR.
Write an equation ${\displaystyle \ln y=mx+b}$ form for the linear relationship shown in the table.
Reading a Book.
$\begin{array}{|cc|}\hline \text{t amount of time reading (hrs.)}& \text{P number of pages left to read}\\ 0& 420\\ 0.25& 405\\ 1.5& 330\\ 3& 240\\ 6& 60\\ \hline\end{array}$
DO NOT USE ANY SPACES between variables, constants, equals signs, and operation signs.
For example DO NOT enter ${\displaystyle y=2x+1}$ DO ENTER: ${\displaystyle y=2x+1}$
Put parentheses around constants that are fractions like this: ${\displaystyle y=\left(\frac{2}{3}\right)x-\left(\frac{1}{2}\right)}$
You may use decimals if the decimal values are exact. (You cannot use $0.33$ for ${\displaystyle \frac{1}{3}}$ because it is not exact, but you can use $0.25$ for ${\displaystyle \frac{1}{4}}$)
To write your equation, use the variables:
${\displaystyle t=}$ amount of time spent reading (hours)
${\displaystyle P\left(t\right)=}$ number of pages left to read.
The equation ${\displaystyle \ln y=mx+b}$ form is _________.

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answer to three decimals.
${\displaystyle g\left(x\right)={\left(\frac{1}{3}\right)}^{x+1};g\left(\frac{1}{2}\right),g\left(\sqrt{2}\right),g(-3.5),g(-1.4)}$