Q1. Solve the absolute value inequality:

Background

Topic: Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values and express the solution in interval notation.

Key Terms and Formulas

• Absolute value: represents the distance of from zero on the number line.

• For , the solution is .

Step-by-Step Guidance

1. Recognize that means the expression inside the absolute value is less than 28 units from zero.

2. Rewrite the inequality as a compound inequality: .

3. Add 7 to all parts of the inequality to isolate .

4. Express the solution in interval notation once you have isolated.

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Q2. Solve and check the linear equation:

Background

Topic: Linear Equations

This question tests your ability to solve linear equations and check your solution by substitution.

Key Terms and Formulas

• Linear equation: An equation of the form .

• Combine like terms and isolate .

Step-by-Step Guidance

1. Expand the parentheses: .

2. Combine like terms to simplify the equation.

3. Isolate by subtracting or adding terms as needed.

4. Once you solve for , substitute back into the original equation to check your answer.

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Q3. Solve the radical equation:

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots and check for extraneous solutions.

Key Terms and Formulas

• Radical equation: An equation with a variable inside a root.

• To solve, isolate the radical and then square both sides.

Step-by-Step Guidance

1. Isolate the square root: .

2. Square both sides to eliminate the radical: .

3. Rearrange the equation to standard quadratic form: .

4. Factor or use the quadratic formula to solve for .

5. Check each solution in the original equation to ensure it is valid (no extraneous solutions).

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Q4. Divide and express in standard form:

Background

Topic: Complex Numbers

This question tests your ability to divide complex numbers and express the result in standard form .

Key Terms and Formulas

• Standard form:

• To divide, multiply numerator and denominator by the conjugate of the denominator.

• Conjugate: For , the conjugate is .

Step-by-Step Guidance

1. Identify the conjugate of the denominator: .

2. Multiply numerator and denominator by the conjugate.

3. Expand and simplify the numerator and denominator.

4. Express the result in the form .

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Q5. After a 90% reduction, you purchase a washing machine on sale for $76. What was the original price?

Background

Topic: Linear Models and Applications

This question tests your ability to set up and solve a percent reduction word problem.

Key Terms and Formulas

• Percent reduction: Sale price = Original price × (1 - reduction rate)

• Let be the original price.

Step-by-Step Guidance

1. Let be the original price.

2. Write the equation: .

3. Solve for by dividing both sides by .

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Q6. Solve using the square root property:

Background

Topic: Square Root Property

This question tests your ability to solve quadratic equations using the square root property.

Key Terms and Formulas

• Square root property: If , then .

Step-by-Step Guidance

1. Take the square root of both sides: .

2. Solve for by adding 8 to both sides.

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Q7. Solve the linear inequality:

Background

Topic: Linear Inequalities

This question tests your ability to solve linear inequalities and express the solution in interval notation.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear expression.

• Combine like terms and isolate .

Step-by-Step Guidance

1. Expand the left side: .

2. Write the inequality: .

3. Move all terms involving to one side and constants to the other.

4. Solve for and express the solution in interval notation.

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Q8. Solve the quadratic equation by factoring:

Background

Topic: Quadratic Equations by Factoring

This question tests your ability to solve quadratic equations by factoring.

Key Terms and Formulas

• Quadratic equation:

• Factoring: Expressing as a product of binomials.

Step-by-Step Guidance

1. Rewrite the equation in standard form: .

2. Factor the quadratic expression.

3. Set each factor equal to zero and solve for .

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Q9. A person will devote 34 years total to sleeping and watching TV over a lifetime. The years spent sleeping exceed the years spent watching TV by 18. How many years will the person spend on each activity?

Background

Topic: Linear Models and Applications

This question tests your ability to set up and solve a system of equations based on a word problem.

Key Terms and Formulas

• Let = years sleeping, = years watching TV.

• System: , .

Step-by-Step Guidance

1. Set up the two equations: and .

2. Substitute from the second equation into the first.

3. Solve for , then use that value to find .

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Q10. Solve the compound inequality:

Background

Topic: Compound Inequalities

This question tests your ability to solve and express compound inequalities in interval notation.

Key Terms and Formulas

• Compound inequality: An inequality involving more than one comparison.

• Combine like terms and isolate .

Step-by-Step Guidance

1. Combine like terms: , so .

2. Add 6 to both sides to isolate .

3. Divide both sides by 2 to solve for .

4. Express the solution in interval notation.

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Q11. Solve using the quadratic formula:

Background

Topic: Quadratic Formula

This question tests your ability to solve quadratic equations using the quadratic formula.

Key Terms and Formulas

• Quadratic formula:

• Standard form:

Step-by-Step Guidance

1. Rewrite the equation in standard form: .

2. Identify , , .

3. Plug these values into the quadratic formula.

4. Simplify under the square root and solve for .

Try solving on your own before revealing the answer!