Q1. Solve the inequality and write the solution set in interval notation:

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.

• Interval notation: A way to describe the set of solutions using intervals.

Step-by-Step Guidance

1. Combine like terms on the left side: .

2. Distribute the $3$ on the right side to both terms inside the parentheses.

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

4. Isolate by dividing both sides by the appropriate coefficient.

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Q2. Solve the inequality and write the solution set in interval notation:

Background

Topic: Linear Inequalities with Variables on Both Sides

This question checks your understanding of how to solve inequalities when the variable appears on both sides.

Key Terms and Formulas:

• Isolate the variable by subtracting from both sides.

• Remember to keep the inequality sign direction unless multiplying/dividing by a negative.

Step-by-Step Guidance

1. Subtract from both sides to eliminate the variable.

2. Simplify the resulting inequality to compare the constants.

3. Write the solution set in interval notation based on the result.

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

Background

Topic: Compound Inequalities

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

Key Terms and Formulas:

• Compound inequality: Two inequalities joined by 'and' (intersection).

• To solve, break into two inequalities and solve each.

Step-by-Step Guidance

1. Break the compound inequality into two parts: and .

2. Solve each inequality for separately.

3. Find the intersection of the two solution sets.

4. Express the final answer in interval notation.

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Q4. Solve the quadratic inequality and write the solution set in interval notation:

Background

Topic: Quadratic Inequalities

This question tests your ability to solve inequalities involving quadratic expressions.

Key Terms and Formulas:

• Quadratic inequality: An inequality involving a quadratic expression.

• Factoring: Expressing the quadratic as a product of binomials.

• Test intervals: Use the zeros to determine where the expression is positive.

Step-by-Step Guidance

1. Set the quadratic equal to zero and factor it to find the critical points.

2. Plot the critical points on a number line to divide it into intervals.

3. Test a value from each interval in the original inequality to see where it holds true.

4. Combine the intervals where the inequality is satisfied.

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Q5. Solve the quadratic inequality and write the solution set in interval notation:

Background

Topic: Quadratic Inequalities

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

Key Terms and Formulas:

• Quadratic inequality: An inequality involving a quadratic expression.

• Move all terms to one side to set the inequality to zero.

• Find the roots and test intervals.

Step-by-Step Guidance

1. Subtract $12$ from both sides to set the inequality to zero.

2. Factor or use the quadratic formula to find the roots.

3. Plot the roots on a number line and test values in each interval.

4. Write the solution set in interval notation for where the inequality holds.

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Q6. Solve the rational inequality and write the solution set in interval notation:

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions.

Key Terms and Formulas:

• Rational inequality: An inequality involving a ratio of polynomials.

• Critical points: Set numerator and denominator to zero to find boundaries.

• Test intervals: Check sign of expression in each interval.

Step-by-Step Guidance

1. Set the numerator and denominator equal to zero to find critical points.

2. Plot these points on a number line to divide it into intervals.

3. Test a value from each interval in the original inequality.

4. Include or exclude endpoints based on the inequality sign and undefined points.

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Q7. Solve the rational inequality and write the solution set in interval notation:

Background

Topic: Rational Inequalities

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

Key Terms and Formulas:

• Rational inequality: An inequality involving a ratio of polynomials.

• Critical points: Set denominator and the expression equal to zero to find boundaries.

• Test intervals: Check sign of expression in each interval.

Step-by-Step Guidance

1. Subtract $5$ from both sides to set the inequality to zero.

2. Combine into a single rational expression.

3. Find the values of that make the numerator or denominator zero.

4. Test intervals between the critical points to determine where the inequality holds.

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Q8. Suppose the velocity of an object is given by , where is time in seconds. Find the intervals where the velocity is negative.

Background

Topic: Quadratic Inequalities (Applications)

This question tests your ability to solve quadratic inequalities in a real-world context.

Key Terms and Formulas:

• Quadratic function:

• Find where (velocity is negative).

• Find roots using the quadratic formula.

Step-by-Step Guidance

1. Set and solve for using the quadratic formula.

2. Identify the two roots, which are the critical points.

3. Test values between the roots to determine where is negative.

4. Express the interval(s) where .

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Q9. Solve the equation:

Background

Topic: Linear Equations

This question tests your ability to solve a simple linear equation for .

Key Terms and Formulas:

• Linear equation: An equation of the form .

• Isolate by performing inverse operations.

Step-by-Step Guidance

1. Subtract $8x$.

2. Divide both sides by to solve for .

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Q10. Solve the equation:

Background

Topic: Linear Equations with Distribution

This question tests your ability to solve a linear equation that requires distributing a negative coefficient.

Key Terms and Formulas:

• Distributive property:

• Isolate by performing inverse operations.

Step-by-Step Guidance

1. Distribute to both terms inside the parentheses.

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

3. Divide both sides by the coefficient of to solve for $x$.

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Q11. Solve:

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation for .

Key Terms and Formulas:

• Linear equation: An equation of the form .

• Isolate the variable by moving all terms to one side.

Step-by-Step Guidance

1. Subtract from both sides to get all terms on one side.

2. Add $5$ to both sides to get all constants on the other side.

3. Divide both sides by the coefficient of to solve for $a$.

Try solving on your own before revealing the answer!