Step-by-Step Guidance for College Algebra Practice Problems

Q1. Solve the equation:

$3x + 7 - 6(x + 1) = 5x + 8$

Background

Topic: Solving Linear Equations

This question tests your ability to solve linear equations by applying the distributive property, combining like terms, and isolating the variable.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

• Combining Like Terms: Add or subtract terms with the same variable.

• Isolating the Variable: Rearranging the equation to solve for $x$.

Step-by-Step Guidance

1. Apply the distributive property to $-6(x + 1)$ to expand the equation.

2. Combine like terms on the left side of the equation.

3. Move all terms containing $x$ to one side and constants to the other side.

4. Simplify the equation to isolate $x$.

Try solving on your own before revealing the answer!

Final Answer: $x = -1$

After distributing, combining like terms, and isolating $x$, you find that $x = -1$.

Q2. Perform the indicated operations:

$(3 - 7i)(4 + i)$

Background

Topic: Complex Numbers (Multiplication)

This question tests your ability to multiply complex numbers using the distributive property (FOIL method) and simplify the result.

Key Terms and Formulas

• Complex Number: A number in the form $a + bi$, where $i$ is the imaginary unit ($i^2 = -1$).

• FOIL Method: Multiply each term in the first binomial by each term in the second binomial.

Step-by-Step Guidance

1. Use the distributive property (FOIL) to expand $(3 - 7i)(4 + i)$.

2. Multiply each term: $3 \times 4$, $3 \times i$, $-7i \times 4$, and $-7i \times i$.

3. Combine the real and imaginary parts.

4. Remember that $i^2 = -1$ and use this to simplify the expression.

Try solving on your own before revealing the answer!

Final Answer: $19 - 25i$

After expanding and simplifying, the product is $19 - 25i$.

Q3. Solve the inequality and write the solution set in interval notation:

$14 \leq 2y + 20 \leq 24$

Background

Topic: Compound Inequalities

This question tests your ability to solve compound (double) inequalities and express the solution in interval notation.

Key Terms and Formulas

• Compound Inequality: An inequality with two comparison symbols, representing a range of values.

• Interval Notation: A way to write the set of solutions using brackets and parentheses.

Step-by-Step Guidance

1. Subtract $20$from all three parts of the inequality to begin isolating$y$.

2. Simplify each part after subtraction.

3. Divide all parts by $2$to solve for$y$.

4. Write the solution in interval notation.

Try solving on your own before revealing the answer!

Final Answer: $[-3, 2]$

The solution set in interval notation is $[-3, 2]$.