Q1. Solve the linear inequality:

Background

Topic: Linear Inequalities

This question tests your ability to solve inequalities involving linear expressions and to express the solution set correctly.

Key Terms and Formulas:

• Linear inequality: An inequality that involves a linear expression (no exponents higher than 1).

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

Step-by-Step Guidance

1. Start by isolating the variable on one side. Subtract from both sides of the inequality.

2. Simplify both sides to combine like terms.

3. Add $7x$.

4. Set up the final step to solve for by dividing both sides by the coefficient of .

Try solving on your own before revealing the answer!

Q2. Find the midpoint of the segment with endpoints and .

Background

Topic: Midpoint Formula

This question tests your understanding of how to find the midpoint between two points in the coordinate plane.

Key Terms and Formulas:

• Midpoint: The point exactly halfway between two given points.

Step-by-Step Guidance

1. Identify the coordinates of the two endpoints: and .

2. Add the -coordinates together: .

3. Add the -coordinates together: .

4. Divide each sum by $2xy$ values of the midpoint.

Try solving on your own before revealing the answer!

Q3. Write the equation of a circle with center and radius $6$.

Background

Topic: Equation of a Circle

This question tests your ability to write the equation of a circle given its center and radius.

Key Terms and Formulas:

• Standard form of a circle:

• is the center, is the radius.

Step-by-Step Guidance

1. Identify the center and the radius from the problem.

2. Substitute , , and into the standard form equation.

3. Simplify the equation by squaring the radius.

Try solving on your own before revealing the answer!

Q4. Solve the absolute value equation:

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute value, which often results in two possible solutions.

Key Terms and Formulas:

• Absolute value: The distance from zero on the number line; always non-negative.

• To solve , set up two equations: and .

Step-by-Step Guidance

1. Set up two separate equations: and .

2. Solve each equation for by first adding $5$ to both sides.

3. Divide both sides by $2x$ in each equation.

Try solving on your own before revealing the answer!

Q5. Given and , find .

Background

Topic: Function Composition

This question tests your understanding of how to compose two functions, meaning to substitute one function into another.

Key Terms and Formulas:

• Composition:

Step-by-Step Guidance

1. Recall that means .

2. Substitute into , so .

3. Expand using the distributive property or binomial expansion.

Try solving on your own before revealing the answer!

Q6. Solve the quadratic equation:

Background

Topic: Solving Quadratic Equations

This question tests your ability to solve a quadratic equation by factoring, completing the square, or using the quadratic formula.

Key Terms and Formulas:

• Quadratic equation:

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

• Quadratic formula:

Step-by-Step Guidance

1. Identify , , and .

2. Look for two numbers that multiply to $6-5$ (for factoring).

3. Set up the factors as where and are the numbers you found.

4. Set each factor equal to zero to solve for .

Try solving on your own before revealing the answer!