Q1. Solve the linear equation for x: 86 = 9x - 4

Background

Topic: Solving Linear Equations

This question tests your ability to solve a basic linear equation for the variable x.

Key Terms and Formulas

• Linear equation: An equation of the form .

• To solve for x, isolate x on one side of the equation.

Step-by-Step Guidance

1. Start by adding 4 to both sides of the equation to move the constant term.

2. After simplifying, you will have an equation where 9x is equal to a number.

3. Divide both sides by 9 to solve for x.

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Q2. Solve for x:

Background

Topic: Solving Linear Equations with Variables on Both Sides

This question tests your ability to simplify and solve linear equations where variables appear on both sides.

Key Terms and Formulas

• Distributive property:

• Combine like terms and isolate x.

Step-by-Step Guidance

1. Distribute the -2 on the right side: .

2. Simplify both sides by combining like terms.

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

4. Solve for x by isolating it.

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Q3. Simplify: and write your answer in the form .

Background

Topic: Complex Numbers (Multiplication)

This question tests your ability to multiply complex numbers and express the result in standard form.

Key Terms and Formulas

• Complex number: , where

• Use the distributive (FOIL) method to multiply.

Step-by-Step Guidance

1. Apply the distributive property (FOIL): Multiply each term in the first parenthesis by each term in the second.

2. Remember that when simplifying.

3. Combine real and imaginary parts to write in form.

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Q4. Factor the polynomial:

Background

Topic: Factoring Difference of Squares

This question tests your ability to recognize and factor expressions of the form .

Key Terms and Formulas

• Difference of squares:

Step-by-Step Guidance

1. Identify and such that and .

2. Write the expression as a difference of squares.

3. Apply the factoring formula for difference of squares.

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Q5. Factor the polynomial:

Background

Topic: Factoring Quadratic Trinomials

This question tests your ability to factor a quadratic expression of the form .

Key Terms and Formulas

• Quadratic trinomial:

• Factoring by grouping or trial and error for integer factors.

Step-by-Step Guidance

1. Multiply and to find the product.

2. Find two numbers that multiply to and add to .

3. Rewrite the middle term using these two numbers and factor by grouping.

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Q6. Factor the polynomial:

Background

Topic: Factoring Perfect Square Trinomials

This question tests your ability to recognize and factor perfect square trinomials.

Key Terms and Formulas

• Perfect square trinomial:

Step-by-Step Guidance

1. Identify and such that and .

2. Check if the middle term is .

3. If so, write the trinomial as a squared binomial.

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Q7. Solve the quadratic equation by factoring:

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to rearrange and factor a quadratic equation to find its solutions.

Key Terms and Formulas

• Quadratic equation:

• Zero product property: If , then or .

Step-by-Step Guidance

1. Move all terms to one side to set the equation to zero.

2. Factor the resulting quadratic expression.

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

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Q8. Solve the quadratic equation by using the square root property:

Background

Topic: Solving Quadratic Equations Using the Square Root Property

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

Key Terms and Formulas

• Square root property: If , then

Step-by-Step Guidance

1. Divide both sides by 4 to isolate .

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

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Q9. Use the quadratic formula to solve:

Background

Topic: Quadratic Formula

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

Key Terms and Formulas

• Quadratic formula:

• For

Step-by-Step Guidance

1. Identify , , and from the equation.

2. Plug these values into the quadratic formula.

3. Calculate the discriminant .

4. Set up the expression for using the formula, but do not compute the final value yet.

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Q10. Solve the rational equation and state what values of x make the denominators zero:

Background

Topic: Rational Equations and Domain Restrictions

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

Key Terms and Formulas

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

• Excluded values: Values that make any denominator zero.

Step-by-Step Guidance

1. Identify values of x that make any denominator zero.

2. Find a common denominator and rewrite the equation without fractions.

3. Solve the resulting linear equation for x.

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Q11. Solve the radical equation and check all proposed solutions:

Background

Topic: Solving Radical Equations

This question tests your ability to solve equations involving square roots and to check for extraneous solutions.

Key Terms and Formulas

• Radical equation: An equation in which the variable is under a radical sign.

• Extraneous solution: A solution that does not satisfy the original equation.

Step-by-Step Guidance

1. Square both sides to eliminate the square root.

2. Rearrange the resulting quadratic equation to standard form.

3. Solve the quadratic equation for x.

4. Check each solution in the original equation to ensure it is valid.

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Q12. Solve the absolute value equation or indicate that the equation has no solution:

Background

Topic: Absolute Value Equations

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

Key Terms and Formulas

• Absolute value: if , if

• To solve , set and .

Step-by-Step Guidance

1. Subtract 7 from both sides to isolate the absolute value.

2. Set up two equations: and .

3. Solve each equation for x.

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Q13. Solve the linear inequality and express the solution in interval notation:

Background

Topic: Linear Inequalities

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

Key Terms and Formulas

• Linear inequality: An inequality involving a linear expression.

• When multiplying or dividing both sides by a negative, reverse the inequality sign.

Step-by-Step Guidance

1. Add to both sides to collect x terms on one side.

2. Simplify and isolate x.

3. Express the solution in interval notation.

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Q14. Solve the absolute value inequality and express the solution in interval notation:

Background

Topic: Absolute Value Inequalities

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

Key Terms and Formulas

• For , the solution is .

Step-by-Step Guidance

1. Add 9 to both sides to isolate the absolute value.

2. Set up the compound inequality based on the absolute value property.

3. Solve for x in the resulting inequality.

4. Express the solution in interval notation.

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Q15 (Extra Credit). Divide and write your answer in the form :

Background

Topic: Division of Complex Numbers

This question tests your ability to divide complex numbers and express the result in standard form.

Key Terms and Formulas

• To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator.

• Conjugate: If , the conjugate is .

Step-by-Step Guidance

1. Multiply numerator and denominator by the conjugate of , which is .

2. Expand both numerator and denominator using distributive property.

3. Simplify the denominator using .

4. Write the result in form.

Try solving on your own before revealing the answer!