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.

Try solving on your own before revealing the answer!

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

Background

Topic: Linear Inequalities

This question checks your understanding of solving inequalities where the variable terms cancel out.

Key Terms and Formulas

• Linear inequality: An inequality with variables to the first power.

• Solution set: The set of all values that satisfy the inequality.

Step-by-Step Guidance

1. Subtract from both sides to eliminate the variable terms.

2. Analyze the resulting inequality involving only constants.

3. Determine if the statement is always true, always false, or sometimes true.

Try solving on your own before revealing the answer!

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' or 'or'.

• Interval notation: A way to write the solution set.

Step-by-Step Guidance

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

2. Solve each part separately for .

3. Find the intersection of the two solution sets.

Try solving on your own before revealing the answer!

Q4. 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.

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

• Test intervals: Checking the sign of the expression in each interval determined by the roots.

Step-by-Step Guidance

1. Rewrite the inequality as .

2. Factor the quadratic expression.

3. Find the roots by setting each factor equal to zero.

4. Use the roots to divide the number line into intervals.

5. Test a value from each interval in the original inequality to determine where the expression is positive.

Try solving on your own before revealing the answer!

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.

• Factoring or quadratic formula: Methods to find the roots.

• Test intervals: Checking the sign of the expression in each interval determined by the roots.

Step-by-Step Guidance

1. Rewrite the inequality as .

2. Factor the quadratic if possible, or use the quadratic formula to find the roots.

3. Use the roots to divide the number line into intervals.

4. Test a value from each interval in the original inequality to determine where the expression is less than or equal to zero.

Try solving on your own before revealing the answer!