Q1. Solve and check the linear equation:

Background

Topic: Solving Linear Equations

This question tests your ability to solve linear equations in one variable by combining like terms and isolating the variable.

Key Terms and Formulas:

• Linear equation: An equation of the form .

• Combine like terms: Add or subtract terms with the same variable.

• Isolate the variable: Get all terms with on one side and constants on the other.

Step-by-Step Guidance

1. Distribute the across on the left side.

2. Combine like terms on the left side to simplify the equation.

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

4. Divide both sides by the coefficient of to solve for .

Try solving on your own before revealing the answer!

Q2. Solve the linear equation:

Background

Topic: Solving Linear Equations

This question tests your ability to solve for when it appears on both sides of the equation.

Key Terms and Formulas:

• Linear equation:

• Isolate by moving all terms to one side and constants to the other.

Step-by-Step Guidance

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

2. Add $10$ to both sides to get all constants on the other side.

3. Simplify both sides and solve for by dividing by the coefficient.

Try solving on your own before revealing the answer!

Q3. Solve the equation:

Background

Topic: Solving Equations with Fractions

This question tests your ability to solve linear equations involving fractions by finding a common denominator.

Key Terms and Formulas:

• Common denominator: The least common multiple of all denominators.

• Clear fractions by multiplying both sides by the common denominator.

Step-by-Step Guidance

1. Identify the least common denominator (LCD) for all terms.

2. Multiply both sides of the equation by the LCD to eliminate fractions.

3. Combine like terms and isolate .

4. Solve for by dividing both sides by the coefficient of .

Try solving on your own before revealing the answer!

Q4. First, write the value(s) that make the denominator(s) zero. Then solve the equation:

Background

Topic: Rational Equations

This question tests your ability to solve rational equations and identify restrictions on the variable.

Key Terms and Formulas:

• Rational equation: An equation involving fractions with variables in the denominator.

• Restrictions: Values that make any denominator zero are excluded from the solution set.

Step-by-Step Guidance

1. Set each denominator equal to zero and solve for to find restrictions.

2. Find the least common denominator (LCD) for all terms.

3. Multiply both sides by the LCD to clear denominators.

4. Solve the resulting linear equation for .

Try solving on your own before revealing the answer!

Q5. First, write the value(s) that make the denominator(s) zero. Then solve the equation:

Background

Topic: Rational Equations

This question tests your ability to solve rational equations and identify excluded values.

Key Terms and Formulas:

• Excluded values: Values that make any denominator zero.

• Clear denominators by multiplying both sides by the LCD.

Step-by-Step Guidance

1. Set each denominator equal to zero and solve for to find restrictions.

2. Find the LCD for all denominators.

3. Multiply both sides by the LCD to eliminate denominators.

4. Solve the resulting equation for and check if any solutions are excluded.

Try solving on your own before revealing the answer!

Q6. Solve the equation:

Background

Topic: Rational Equations

This question tests your ability to solve rational equations with quadratic denominators.

Key Terms and Formulas:

• Factor as .

• Find the LCD and multiply both sides to clear denominators.

Step-by-Step Guidance

1. Factor as .

2. Identify the LCD: .

3. Multiply both sides by the LCD to clear denominators.

4. Solve the resulting linear equation for .

Try solving on your own before revealing the answer!

Q7. Solve the equation:

Background

Topic: Rational Equations

This question tests your ability to solve rational equations with multiple rational expressions.

Key Terms and Formulas:

• Factor denominators where possible.

• Find the LCD and multiply both sides to clear denominators.

Step-by-Step Guidance

1. Factor and as and , respectively.

2. Identify the LCD: .

3. Multiply both sides by the LCD to clear denominators.

4. Solve the resulting linear equation for .

Try solving on your own before revealing the answer!

Q8. Determine whether the equation is an identity, a conditional equation, or an inconsistent equation:

Background

Topic: Types of Equations

This question tests your ability to classify equations as identities, conditional, or inconsistent.

Key Terms and Formulas:

• Identity: True for all values of the variable.

• Conditional: True for some values of the variable.

• Inconsistent: Not true for any value of the variable.

Step-by-Step Guidance

1. Distribute and simplify both sides of the equation.

2. Combine like terms and move all terms to one side.

3. Check if the resulting equation is always true, never true, or true for some values.

Try classifying on your own before revealing the answer!

Q9. Determine whether the equation is an identity, a conditional equation, or an inconsistent equation:

Background

Topic: Types of Equations

This question tests your ability to classify equations as identities, conditional, or inconsistent.

Key Terms and Formulas:

• Identity: True for all values of the variable.

• Conditional: True for some values of the variable.

• Inconsistent: Not true for any value of the variable.

Step-by-Step Guidance

1. Expand and simplify both sides of the equation.

2. Combine like terms and move all terms to one side.

3. Check if the resulting equation is always true, never true, or true for some values.

Try classifying on your own before revealing the answer!

Q10. Solve the equation by factoring:

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to factor quadratic equations and solve for .

Key Terms and Formulas:

• Quadratic equation:

• Factoring: Expressing the quadratic as

• Zero product property: If , then or

Step-by-Step Guidance

1. Factor into two binomials.

2. Set each factor equal to zero.

3. Solve each resulting linear equation for .

Try factoring and solving before revealing the answer!

Q11. Solve the equation by the square root property:

Background

Topic: Solving Quadratic Equations by Square Root Property

This question tests your ability to isolate and use the square root property.

Key Terms and Formulas:

• Square root property: If , then

Step-by-Step Guidance

1. Subtract $5x^2$ term.

2. Divide both sides by $4x^2$.

3. Take the square root of both sides, remembering to include both the positive and negative roots.

Try solving before revealing the answer!

Q12. Solve the equation by the square root property:

Background

Topic: Solving Quadratic Equations by Square Root Property

This question tests your ability to use the square root property on equations of the form .

Key Terms and Formulas:

• Square root property: If , then

Step-by-Step Guidance

1. Take the square root of both sides, remembering to include both positive and negative roots.

2. Solve the resulting linear equations for .

Try solving before revealing the answer!

Q13. Claire has received scores of 85, 88, 87, and 80 on her algebra tests. What score must she receive on the fifth test to have an overall test score average of at least 82?

Background

Topic: Averages and Inequalities

This question tests your ability to set up and solve an inequality involving averages.

Key Terms and Formulas:

• Average:

• Set up an inequality to represent "at least 82".

Step-by-Step Guidance

1. Let be the score on the fifth test.

2. Write the inequality: .

3. Multiply both sides by $5$ to clear the denominator.

4. Solve for to find the minimum score needed.

Try setting up and solving before revealing the answer!

Q14. Solve the linear inequality and express the solution set in interval notation:

Background

Topic: Solving Linear Inequalities with Fractions

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

Key Terms and Formulas:

• Find the least common denominator (LCD) to clear fractions.

• Isolate and solve the inequality.

• Express the solution in interval notation.

Step-by-Step Guidance

1. Find the LCD for all denominators (8, 12, 24).

2. Multiply both sides by the LCD to eliminate fractions.

3. Combine like terms and isolate .

4. Express the solution in interval notation.

Try solving before revealing the answer!

Q15. Solve the linear inequality:

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas:

• Distribute and combine like terms.

• Isolate and solve the inequality.

• Express the solution in interval notation.

Step-by-Step Guidance

1. Distribute the $4$ on the right side.

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

3. Solve for and write the solution in interval notation.

Try solving before revealing the answer!

Q16. Solve the linear inequality:

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas:

• Distribute and combine like terms.

• Isolate and solve the inequality.

• Express the solution in interval notation.

Step-by-Step Guidance

1. Distribute the $2$ on the right side.

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

3. Solve for and write the solution in interval notation.

Try solving before revealing the answer!

Q17. Solve the equation using the quadratic formula:

Background

Topic: Quadratic Formula

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

Key Terms and Formulas:

• Quadratic formula:

• For

Step-by-Step Guidance

1. Identify , , .

2. Plug these values into the quadratic formula.

3. Calculate the discriminant .

4. Set up the two possible solutions using .

Try solving before revealing the answer!