Improve Your Algebra Skills with Practice Problems

Solve: ${\displaystyle \frac{1}{2}(\frac{x}{3}+1)=\frac{x}{4}-2}$

Simplify ${\displaystyle \frac{{\cos }^2\left(t\right)}{1-{\cos }^2\left(t\right)}}$ to an expression involving a single trig function with no fractions.

"Perform the indicated operations in the figure below.
${\displaystyle 17\frac{2}{5}+30\frac{2}{3}}$

Part A: Find the local linear approximation of the function ${\displaystyle f\left(x\right)=\sqrt{3+x}\text{ at }{x}_0=6}$,
Part B use part A to approximate ${\displaystyle \sqrt{8.9}\text{ as }\sqrt{9.1}}$. Answers must be in fractions

How can I understand Equations better?

How many permutations of the letters ABCDEFG contain the strings ABC and DE?

Evaluate
${\displaystyle \frac{8}{9}+\frac{15}{2}}$

Find the LCD of the following expressions. Write LCD in factored form.
${\displaystyle \frac{5}{16p^2+8p+1}}$ and ${\displaystyle \frac{4p}{-8p^2+10p+3}}$
LCD: ?

Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients.
a) ${\displaystyle \frac{x^4+6}{x^5+7x^3}}$
b) ${\displaystyle \frac{2}{{(x^2-9)}^2}}$

Set-up and solve the following algebraically:
A group of 7 students go out for lunch. If two students have hamburgers and five students have hot dogs, the bill will be $18.75. If five students have hamburgers and two students have hot dogs, the bill will be $23.25. What is the price of a hamburger?

For exercise, solve the equations and inequalities. Write the solutions sets to the inequalities in interval notation if possible.
${\displaystyle \frac{3-x}{x+5}\ge 1}$

Whether the given statement is true or false. If false, then to provide the correct statement.
Given: The statement "The inequalities ${\displaystyle (x+3)(x-1)\ge 0\text{and}\frac{x+3}{x-1}\ge 0}$ have the same solution set."

Solve the exponential equation ${\displaystyle 3^{2x}+3^x-2=0.}$ Express the solution set in terms of natural logarithms or common logarithms.Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.

Divide the fractions and mixed numbers
${\displaystyle \frac{\frac{54}{9}}{\frac{82}{3}}}$

Give the 4 cases of partial fractions with examples.

Subtraction problem
${\displaystyle \frac{x}{9}-\frac{x+2}{9}}$

Consider the function ${\displaystyle f\left(x\right)=3\sin \left(\frac{\pi }{3}(x-4)\right)+5}$.
State the amplitude A, period P, and midline. State the phase shift and vertical translation. In the full period [0, P], state the maximum and minimum y-values and their corresponding x-values.
Enter the exact answers.
Amplitude: ${\displaystyle A=}$
Period: ${\displaystyle P=}$
Midline: ${\displaystyle y=}$
The phase shift is:
a. up 4 units
b. 4 units to the right
c. 4 units to the left
d. 5 units to the left
The vertical translation is:
a. up 4 units
b. down 5 units
c. up 5 units
d. down 4 units
Hints for the maximum and minimum values of f(x):
The maximum value of ${\displaystyle y=\sin \left(x\right)}$ is ${\displaystyle y=1}$ and the corresponding x values are ${\displaystyle x=\frac{\pi }{2}}$ and multiples of ${\displaystyle 2\pi }$ less than and more than this x value. You may want to solve ${\displaystyle \frac{\pi }{3}(x-4)=\frac{\pi }{2}}$.
The minimum value of ${\displaystyle y=\sin \left(x\right)}$ is ${\displaystyle y=-1}$ and the corresponding x values are ${\displaystyle x=\frac{3\pi }{2}}$ and multiples of ${\displaystyle 2\pi }$ less than and more than this x value. You may want to solve ${\displaystyle \frac{\pi }{3}(x-4)=\frac{3\pi }{2}}$.
If you get a value for x that is less than 0, you could add multiples of P to get into the next cycles.
If you get a value for x that is more than P, you could subtract multiples of P to get into the previous cycles.
For x in the interval [0, P], the maximum y-value and corresponding x-value is at:
x=
y=
For x in the interval [0, P], the minimum y-value and corresponding x-value is at:
${\displaystyle x=}$
${\displaystyle y=}$

A club has 25 members. a) How many ways are there to choose four members of the club to serve on an executive committee? b) How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?

How many permutations of the letters ABCDEFG contain the string CFGA?