Q1. Perform the calculation:

Background

Topic: Real Number Operations (Addition)

This question tests your understanding of how to add integers, especially when dealing with negative and positive numbers.

Key Terms:

• Integer: A whole number that can be positive, negative, or zero.

• Addition of integers: When adding a negative and a positive number, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.

Step-by-Step Guidance

1. Identify the two numbers: and $30$.

2. Since the numbers have different signs, subtract the absolute values: .

3. Determine the sign of the result based on which number has the larger absolute value.

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Q2. Perform the calculation:

Background

Topic: Real Number Operations (Subtraction)

This question tests your ability to subtract integers, especially when both numbers are negative.

Key Terms:

• Subtraction of integers: Subtracting a positive number from a negative number is the same as adding two negative numbers.

Step-by-Step Guidance

1. Rewrite the subtraction as addition: .

2. Add the two negative numbers together by adding their absolute values and keeping the negative sign.

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Q3. Perform the calculation:

Background

Topic: Real Number Operations (Multiplication)

This question tests your understanding of multiplying two negative numbers.

Key Terms:

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

Step-by-Step Guidance

1. Multiply the absolute values: .

2. Since both numbers are negative, the result will be positive.

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Q4. Perform the calculation:

Background

Topic: Real Number Operations (Division)

This question tests your ability to divide a negative number by a positive number.

Key Terms:

• Division of integers: When dividing numbers with different signs, the result is negative.

Step-by-Step Guidance

1. Divide the absolute values: .

2. Since the dividend is negative and the divisor is positive, the result will be negative.

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Q5. Evaluate for

Background

Topic: Evaluating Algebraic Expressions

This question tests your ability to substitute a value for a variable and simplify the expression.

Key Terms:

• Substitution: Replacing the variable with the given value.

• Order of operations: Follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Step-by-Step Guidance

1. Substitute into the expression: becomes .

2. Simplify the double negative: becomes $5$.

3. Add to complete the evaluation.

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Q6. Evaluate for

Background

Topic: Evaluating Algebraic Expressions with Exponents

This question tests your ability to substitute a value for a variable, handle exponents, and simplify the result.

Key Terms and Formulas:

• Exponentiation: means multiplied by itself.

• Substitution: Replace with in the expression.

Step-by-Step Guidance

1. Substitute into the expression: becomes .

2. Calculate first, then multiply by $2$.

3. Multiply by .

4. Combine all terms to simplify the expression.

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Q7. Combine like terms:

Background

Topic: Simplifying Algebraic Expressions

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

Key Terms:

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

• Combine: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Group the terms: and .

2. Group the terms: and .

3. Group the constants: and $10$.

4. Add or subtract the coefficients for each group to combine like terms.

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Q8. Remove parentheses and combine like terms:

Background

Topic: Simplifying Algebraic Expressions (Distributive Property)

This question tests your ability to use the distributive property and combine like terms.

Key Terms and Formulas:

• Distributive Property:

• Like terms: Combine terms with the same variable.

Step-by-Step Guidance

1. Apply the distributive property: and .

2. Write the expression without parentheses.

3. Combine like terms: terms and constants.

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Q9. Remove parentheses and combine like terms:

Background

Topic: Simplifying Algebraic Expressions (Parentheses and Like Terms)

This question tests your ability to remove parentheses, especially when a negative sign is involved, and combine like terms.

Key Terms:

• Negative sign before parentheses: Distribute the negative to each term inside.

• Like terms: Combine terms with the same variable.

Step-by-Step Guidance

1. Distribute the negative sign: becomes .

2. Rewrite the expression: .

3. Combine like terms: and .

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Q10. Remove parentheses and combine like terms:

Background

Topic: Simplifying Algebraic Expressions (Distributive Property and Like Terms)

This question tests your ability to use the distributive property, remove parentheses, and combine like terms.

Key Terms and Formulas:

• Distributive Property:

• Like terms: Combine terms with the same variable.

Step-by-Step Guidance

1. Apply the distributive property: and .

2. Rewrite the expression without parentheses.

3. Combine like terms: terms and constants.

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Q11. Solve the equation:

Background

Topic: Solving Linear Equations (Addition Principle)

This question tests your ability to isolate the variable using the addition principle.

Key Terms and Formulas:

• Addition Principle: You can add the same number to both sides of an equation without changing the solution.

Step-by-Step Guidance

1. Identify the variable term () and the constant () on the left side.

2. Add $2x$.

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Q12. Solve the equation:

Background

Topic: Solving Linear Equations (Subtraction Principle)

This question tests your ability to isolate the variable using the subtraction principle.

Key Terms and Formulas:

• Subtraction Principle: You can subtract the same number from both sides of an equation without changing the solution.

Step-by-Step Guidance

1. Identify the variable term () and the constant () on the left side.

2. Subtract $2x$.

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Q13. Solve the equation:

Background

Topic: Solving Linear Equations (Multiplication/Division Principle)

This question tests your ability to isolate the variable using division.

Key Terms and Formulas:

• Division Principle: You can divide both sides of an equation by the same nonzero number.

Step-by-Step Guidance

1. Identify the coefficient of (which is $2$).

2. Divide both sides of the equation by $2x$.

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Q14. Solve the equation:

Background

Topic: Solving Linear Equations with Fractions

This question tests your ability to solve equations where the variable is multiplied by a fraction.

Key Terms and Formulas:

• Multiplicative Inverse: To undo multiplication by a fraction, multiply both sides by its reciprocal.

Step-by-Step Guidance

1. Identify the coefficient of ().

2. Multiply both sides of the equation by $2\frac{1}{2}x$.

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