Q1. Solve the absolute value inequality: |7x - 7| < 28. Express the solution set in interval notation.

Background

Topic: Absolute Value Inequalities

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

Key Terms and Formulas:

• Absolute value inequality: is equivalent to .

• Interval notation: A way to represent solution sets on the real number line.

Step-by-Step Guidance

1. Rewrite the inequality as a compound inequality: .

2. Add 7 to all three parts of the inequality to isolate the term.

3. Divide all parts of the inequality by 7 to solve for .

4. Express the solution in interval notation.

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Q2. Solve for :

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation involving fractions and variables on both sides.

Key Terms and Formulas:

• Linear equation: An equation of the form .

• To solve, isolate on one side.

Step-by-Step Guidance

1. Start with .

2. Multiply both sides by to make the coefficients positive.

3. Multiply both sides by 2 to clear the fraction.

4. Combine like terms to isolate .

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Q3. Solve and check the linear equation:

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation with parentheses and to check your solution.

Key Terms and Formulas:

• Distributive property: .

• Combine like terms and isolate .

Step-by-Step Guidance

1. Expand the equation: .

2. Distribute the negative sign: .

3. Combine like terms to simplify the equation.

4. Isolate by subtracting or adding as needed.

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Q4. Solve the radical equation:

Background

Topic: Radical Equations

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

Key Terms and Formulas:

• Radical equation: An equation in which the variable is under a radical sign.

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

Step-by-Step Guidance

1. Start with .

2. Square both sides to eliminate the square root.

3. Rearrange the resulting quadratic equation to standard form.

4. Solve the quadratic equation for .

5. Check all proposed solutions in the original equation to rule out extraneous solutions.

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Q5. Solve the formula for :

Background

Topic: Literal Equations

This question tests your ability to solve for a specified variable in a formula.

Key Terms and Formulas:

• Literal equation: An equation involving two or more variables.

• To solve for , isolate $q$ on one side.

Step-by-Step Guidance

1. Start with .

2. Subtract from both sides to isolate the term.

3. Divide both sides by to solve for .

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Q6. Divide and express the result in standard form:

Background

Topic: Complex Numbers

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

Key Terms and Formulas:

• Standard form: , where and are real numbers.

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

Step-by-Step Guidance

1. Identify the conjugate of the denominator: .

2. Multiply numerator and denominator by the conjugate.

3. Expand both numerator and denominator using distributive property.

4. Simplify the denominator using .

5. Write the result in the form .

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Q7. After a 90% reduction, you purchase a new washing machine on sale for $76. What was the original price?

Background

Topic: Percentages and Reverse Percent Problems

This question tests your ability to work backwards from a percentage reduction to find the original price.

Key Terms and Formulas:

• Let be the original price. After a 90% reduction, you pay 10% of the original price.

• Equation:

Step-by-Step Guidance

1. Set up the equation: .

2. Solve for by dividing both sides by .

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Q8. Solve the equation by the square root property:

Background

Topic: Quadratic Equations (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. Start with .

2. Take the square root of both sides, remembering to use both the positive and negative roots.

3. Solve for by adding 8 to both sides.

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Q9. Solve the inequality: . Express the solution in interval notation and choose the correct graph.

Background

Topic: Linear Inequalities

This question tests your ability to solve linear inequalities and represent the solution on a number line.

Key Terms and Formulas:

• Distributive property: .

• Combine like terms and isolate .

• Interval notation for inequalities.

Step-by-Step Guidance

1. Expand the left side: .

2. Combine like terms and move all terms to one side and constants to the other.

3. Solve for by dividing both sides by the appropriate coefficient.

4. Express the solution in interval notation.

5. Match the solution to the correct graph.

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Q10. Solve the absolute value equation:

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute values.

Key Terms and Formulas:

• Absolute value equation: is equivalent to or .

Step-by-Step Guidance

1. Set up two equations: and .

2. Solve each equation for .

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Q11. Solve the equation by factoring:

Background

Topic: Quadratic Equations (Factoring)

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

Key Terms and Formulas:

• Standard form: .

• Factoring quadratics: Find two numbers that multiply to and add to .

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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Q12. According to statistics, a person will devote 34 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 18. Over the lifetime, how many years will the person spend on each activity?

Background

Topic: Systems of Linear Equations (Word Problems)

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 watching TV, = years sleeping.

• Equations: , .

Step-by-Step Guidance

1. Write the two equations based on the problem statement.

2. Substitute into the first equation.

3. Solve for (years watching TV).

4. Use the value of to find (years sleeping).

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Q13. Solve the compound inequality: . Express the solution in interval notation.

Background

Topic: Compound Inequalities

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

Key Terms and Formulas:

• Compound inequality: Two inequalities joined by 'and'.

• To solve, isolate in the middle.

Step-by-Step Guidance

1. Start with .

2. Subtract 1 from all three parts of the inequality.

3. Divide all parts by 2 to solve for .

4. Express the solution in interval notation.

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Q14. Solve the equation using the quadratic formula:

Background

Topic: Quadratic Equations (Quadratic Formula)

This question tests your ability to use the quadratic formula to solve a quadratic equation.

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 set up the two possible solutions.

Try solving on your own before revealing the answer!