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College Algebra Practice Exam Guidance

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Q1. Solve the equation

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation for by isolating the variable.

Key Terms and Formulas:

  • Linear equation: An equation of the form

  • Solving for : Rearranging terms to isolate $x$

Step-by-Step Guidance

  1. Move all terms involving to one side and constants to the other. For example, subtract from both sides.

  2. Combine like terms to simplify the equation.

  3. Isolate by dividing both sides by the coefficient of $x$.

Try solving on your own before revealing the answer!

Q2. Solve

Background

Topic: Linear Equations

This question tests your ability to simplify and solve linear equations, including combining like terms and distributing.

Key Terms and Formulas:

  • Distributive property:

  • Combining like terms

Step-by-Step Guidance

  1. Expand using the distributive property.

  2. Combine like terms on both sides of the equation.

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

Try solving on your own before revealing the answer!

Q3. Solve

Background

Topic: Quadratic Equations

This question tests your ability to solve a quadratic equation by factoring.

Key Terms and Formulas:

  • Quadratic equation:

  • Factoring: Expressing as

Step-by-Step Guidance

  1. Identify , , .

  2. Look for two numbers that multiply to $10-7$.

  3. Write the equation in factored form and set each factor equal to zero.

Try solving on your own before revealing the answer!

Q4. If the discriminant of a quadratic equation is $49$, how many solutions does it have?

Background

Topic: Quadratic Formula and Discriminant

This question tests your understanding of the discriminant and how it determines the number of real solutions to a quadratic equation.

Key Terms and Formulas:

  • Discriminant:

  • If , two real solutions; , one real solution; , no real solutions.

Step-by-Step Guidance

  1. Recall the meaning of the discriminant in the quadratic formula.

  2. Determine what implies about the number of real solutions.

Try solving on your own before revealing the answer!

Q5. Solve

Background

Topic: Rational Exponents

This question tests your ability to solve equations involving rational exponents.

Key Terms and Formulas:

  • Rational exponent:

  • Inverse operations: Use roots and powers to isolate .

Step-by-Step Guidance

  1. Rewrite $27$ as a power if possible.

  2. Apply the inverse operation to both sides to isolate .

  3. Solve for by subtracting $1$ from both sides.

Try solving on your own before revealing the answer!

Q6. How many solutions does have?

Background

Topic: Zero Product Property

This question tests your ability to use the zero product property to find all real solutions.

Key Terms and Formulas:

  • Zero product property: If , then or .

Step-by-Step Guidance

  1. Set each factor equal to zero: and .

  2. Solve each equation for .

  3. Count the total number of distinct solutions.

Try solving on your own before revealing the answer!

Q7. Find the solution set to the compound inequality or

Background

Topic: Compound Inequalities

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

Key Terms and Formulas:

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

  • Interval notation

Step-by-Step Guidance

  1. Solve each inequality separately for .

  2. Combine the solution sets using 'or' (union).

  3. Express the final solution in interval notation.

Try solving on your own before revealing the answer!

Q8. Find the solution set to the compound inequality

Background

Topic: Compound Inequalities

This question tests your ability to solve a double inequality and express the solution set in interval notation.

Key Terms and Formulas:

  • Double inequality:

  • Interval notation

Step-by-Step Guidance

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

  2. Solve each part for .

  3. Find the intersection of the two solution sets.

Try solving on your own before revealing the answer!

Q9. Solve

Background

Topic: Absolute Value Inequalities

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

Key Terms and Formulas:

  • Absolute value inequality: means or

Step-by-Step Guidance

  1. Set up two inequalities: and .

  2. Solve each inequality for .

  3. Express the solution as a union of intervals.

Try solving on your own before revealing the answer!

Q10. Solve

Background

Topic: Absolute Value Inequalities

This question tests your understanding of absolute value inequalities, especially when the right side is negative.

Key Terms and Formulas:

  • Absolute value:

  • Note: Absolute value is always non-negative

Step-by-Step Guidance

  1. Consider what it means for to be greater than or equal to a negative number.

  2. Recall that is always non-negative.

  3. Determine the solution set based on this property.

Try solving on your own before revealing the answer!

Q11(a). Solve

Background

Topic: Polynomial Equations

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

Key Terms and Formulas:

  • Factoring:

  • Zero product property

Step-by-Step Guidance

  1. Move all terms to one side: .

  2. Factor out the greatest common factor.

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

Try solving on your own before revealing the answer!

Q11(b). Solve

Background

Topic: Radical Equations

This question tests your ability to solve equations involving square roots.

Key Terms and Formulas:

  • Radical equation: Contains a square root

  • Isolate the radical and square both sides

Step-by-Step Guidance

  1. Isolate the square root on one side.

  2. Square both sides to eliminate the radical.

  3. Solve the resulting quadratic equation for .

Try solving on your own before revealing the answer!

Q11(c). Solve

Background

Topic: Quadratic Equations in Disguise

This question tests your ability to recognize and solve a quadratic equation by substitution.

Key Terms and Formulas:

  • Let , then substitute into the equation

  • Quadratic formula:

Step-by-Step Guidance

  1. Let and rewrite the equation in terms of .

  2. Solve the quadratic equation for .

  3. Substitute back to find .

Try solving on your own before revealing the answer!

Q11(d). Solve

Background

Topic: Linear Equations

This question tests your ability to solve linear equations with distribution and combining like terms.

Key Terms and Formulas:

  • Distributive property

  • Combining like terms

Step-by-Step Guidance

  1. Expand using the distributive property.

  2. Combine like terms on both sides.

  3. Isolate and solve.

Try solving on your own before revealing the answer!

Q11(d). Solve

Background

Topic: Quadratic Equations

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

Key Terms and Formulas:

  • Quadratic formula:

Step-by-Step Guidance

  1. Identify , , .

  2. Plug these values into the quadratic formula.

  3. Simplify under the square root and continue solving for .

Try solving on your own before revealing the answer!

Q11(e). Solve

Background

Topic: Absolute Value Equations

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

Key Terms and Formulas:

  • Absolute value: means or

Step-by-Step Guidance

  1. Set up two equations: and .

  2. Solve each equation for .

Try solving on your own before revealing the answer!

Q11(f). Solve

Background

Topic: Absolute Value Equations

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

Key Terms and Formulas:

  • Absolute value: means or

Step-by-Step Guidance

  1. Set up two equations: and .

  2. Solve each equation for .

Try solving on your own before revealing the answer!

Q12. Solve the inequality using the test-point method. Write your answer in interval notation.

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities using the test-point method and express the solution in interval notation.

Key Terms and Formulas:

  • Quadratic inequality

  • Test-point method: Check values in intervals determined by the roots

Step-by-Step Guidance

  1. Find the roots of .

  2. Divide the real line into intervals based on these roots.

  3. Test a value from each interval to see where the inequality holds.

Try solving on your own before revealing the answer!

Q13(a). Solve for in the formula

Background

Topic: Solving for a Variable

This question tests your ability to rearrange a formula to solve for a specific variable.

Key Terms and Formulas:

  • Linear equation:

  • Solving for

Step-by-Step Guidance

  1. Subtract from both sides to isolate .

  2. Divide both sides by to solve for .

Try solving on your own before revealing the answer!

Q13(b). For the above formula, if , , and , what is ?

Background

Topic: Substitution in Formulas

This question tests your ability to substitute values into a formula and solve for the unknown variable.

Key Terms and Formulas:

  • Substitute values into

  • Solve for

Step-by-Step Guidance

  1. Plug , , and into the formula.

  2. Subtract from and divide by to solve for .

Try solving on your own before revealing the answer!

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