Step-by-Step Guidance for Beginning Algebra Test (Module 1: 1.2–1.8)

Q1. Which sets does the number 419 belong to? (Natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers)

Background

Topic: Number Sets

This question tests your understanding of the different sets of numbers in mathematics and how a specific number fits into these categories.

Key Terms:

• Natural numbers: Counting numbers starting from 1 (1, 2, 3, ...)

• Whole numbers: Natural numbers plus 0 (0, 1, 2, 3, ...)

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...)

• Rational numbers: Numbers that can be written as a fraction of two integers

• Irrational numbers: Numbers that cannot be written as a fraction (like $\sqrt{2}$, $\pi$)

• Real numbers: All rational and irrational numbers

Step-by-Step Guidance

1. Consider whether 419 is a counting number (natural number).

2. Check if 419 is a whole number (does it include 0 or is it positive?).

3. Determine if 419 is an integer (is it a whole number, positive or negative?).

4. Ask if 419 can be written as a fraction (rational number).

5. Decide if 419 is irrational (can it NOT be written as a fraction?).

6. Finally, see if 419 is a real number (is it on the number line?).

Try solving on your own before revealing the answer!

Q2. Simplify the expression: 7g - 3 - 7 - 7g

Background

Topic: Combining Like Terms

This question tests your ability to combine like terms in an algebraic expression.

Key Terms:

• Like terms: Terms that have the same variable raised to the same power.

Step-by-Step Guidance

1. Identify the like terms in the expression (terms with 'g' and constant terms).

2. Combine the coefficients of the 'g' terms: $7g - 7g$.

3. Combine the constant terms: $-3 - 7$.

4. Write the simplified expression by adding the results from steps 2 and 3.

Try solving on your own before revealing the answer!

Q3. Divide: 40 \div (-8)

Background

Topic: Division of Integers

This question tests your understanding of dividing positive and negative numbers.

Key Terms:

• Integer division: Dividing one whole number by another.

• Sign rules: A positive divided by a negative is negative.

Step-by-Step Guidance

1. Write the division as a fraction: $\frac{40}{-8}$.

2. Divide the absolute values: $\frac{40}{8}$.

3. Apply the sign rule: since you are dividing a positive by a negative, the result is negative.

Try solving on your own before revealing the answer!

Q4. Simplify the expression: 42 \div 7 \cdot 2

Background

Topic: Order of Operations

This question tests your ability to apply the correct order of operations (PEMDAS/BODMAS) to simplify an expression.

Key Terms:

• Order of operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Step-by-Step Guidance

1. Identify the operations: division and multiplication.

2. According to the order of operations, perform division and multiplication from left to right.

3. First, divide 42 by 7.

4. Then, multiply the result by 2.

Try solving on your own before revealing the answer!

Q5. Multiply: -2 \times (-6)

Background

Topic: Multiplication of Integers

This question tests your understanding of multiplying negative numbers.

Key Terms:

• Multiplication rules: The product of two negative numbers is positive.

Step-by-Step Guidance

1. Multiply the absolute values: $2 \times 6$.

2. Determine the sign: negative times negative equals positive.

3. Write the final product with the correct sign.

Try solving on your own before revealing the answer!

Q6. Evaluate the expression when $x = -3$ and $y = -2$: $\frac{x^2 + y}{2y}$

Background

Topic: Substitution and Order of Operations

This question tests your ability to substitute values into an expression and simplify using order of operations.

Key Terms and Formulas:

• Substitution: Replacing variables with given values.

• Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Step-by-Step Guidance

1. Substitute $x = -3$ and $y = -2$ into the expression: $\frac{(-3)^2 + (-2)}{2 \times (-2)}$.

2. Calculate $(-3)^2$.

3. Add the result to $-2$ in the numerator.

4. Multiply $2 \times (-2)$ in the denominator.

5. Write the fraction with the simplified numerator and denominator.

Try solving on your own before revealing the answer!