College Algebra Practice Quiz Guidance

Q1. Solve: \frac{5}{6}(x - \frac{3}{4}) - \frac{2}{3}x = \frac{7}{8}(x + \frac{5}{6}) - \frac{1}{4}

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve linear equations involving fractions and variable expressions. You'll need to combine like terms and isolate the variable.

Key Terms and Formulas

• Linear equation: An equation of the form $ax + b = cx + d$.

• Combine like terms: Group terms with the same variable.

• Clear fractions: Multiply both sides by the least common denominator (LCD) to eliminate fractions.

Step-by-Step Guidance

1. Expand each term: Distribute the fractions across the parentheses.

2. Write out the expanded equation: $\frac{5}{6}x - \frac{5}{6} \cdot \frac{3}{4} - \frac{2}{3}x = \frac{7}{8}x + \frac{7}{8} \cdot \frac{5}{6} - \frac{1}{4}$

3. Simplify the constants: Calculate $\frac{5}{6} \cdot \frac{3}{4}$ and $\frac{7}{8} \cdot \frac{5}{6}$.

4. Combine like terms: Group all $x$ terms on one side and constants on the other.

5. Multiply both sides by the LCD to clear fractions (find the LCD of all denominators).

Try solving on your own before revealing the answer!

Q2. Find the zeroes of $g(x) = \frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x$

Background

Topic: Finding Zeroes of Linear Functions

This question asks you to find the values of $x$ that make $g(x) = 0$. This is a fundamental skill in algebra, often used to solve equations and analyze functions.

Key Terms and Formulas

• Zeroes: Values of $x$ where $g(x) = 0$.

• Linear function: $g(x) = ax + b$.

Step-by-Step Guidance

1. Set $g(x)$ equal to zero: $\frac{7}{9}x - \frac{5}{4} - \frac{2}{3}x = 0$

2. Combine like terms: Group the $x$ terms together.

3. Simplify the coefficients: Find a common denominator for $\frac{7}{9}$ and $-\frac{2}{3}$.

4. Isolate $x$ by moving the constant to the other side.

5. Solve for $x$ by dividing both sides by the coefficient of $x$.

Try solving on your own before revealing the answer!

Q3. Find the domain of the function $f(x) = \sqrt{12 - 3x - 2}$ and write it in interval notation.

Background

Topic: Domain of Radical Functions

This question tests your understanding of how to find the domain of a function involving a square root. The expression inside the square root must be non-negative.

Key Terms and Formulas

• Domain: The set of all possible input values ($x$) for which the function is defined.

• Square root function: $f(x) = \sqrt{A}$ is defined only when $A \geq 0$.

Step-by-Step Guidance

1. Set the expression inside the square root greater than or equal to zero: $12 - 3x - 2 \geq 0$

2. Simplify the inequality: Combine like terms to get $10 - 3x \geq 0$

3. Solve for $x$: Move $3x$ to the other side and divide by 3.

4. Write the solution in interval notation.

Try solving on your own before revealing the answer!