College Algebra Practice Set I – Step-by-Step Guidance

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Q1. Solve: \( \frac{x}{2} = 3x^4 + 5 \)

Background

Topic: Solving Polynomial Equations

This question tests your ability to solve equations involving polynomials and rational expressions.

Key Terms and Formulas:

Polynomial equation: An equation involving terms with variables raised to whole number powers.
To solve, isolate the variable and set the equation to zero if possible.

Step-by-Step Guidance

Multiply both sides of the equation by 2 to eliminate the denominator.
Rearrange the equation so that all terms are on one side, resulting in a polynomial equation set equal to zero.
Combine like terms and write the equation in standard form (highest degree to lowest degree).
Look for possible factoring or use the Rational Root Theorem to find solutions.

Try solving on your own before revealing the answer!

Q2. Solve: \( \frac{5}{2x} - \frac{8}{9} = \frac{1}{18} - \frac{1}{3x} \)

Background

Topic: Solving Rational Equations

This question tests your ability to solve equations involving rational expressions (fractions with variables in the denominator).

Key Terms and Formulas:

Rational equation: An equation containing one or more rational expressions.
To solve, find a common denominator to combine terms and isolate the variable.

Step-by-Step Guidance

Identify the least common denominator (LCD) for all terms in the equation.
Multiply both sides of the equation by the LCD to clear all denominators.
Simplify the resulting equation and collect like terms.
Solve for the variable using algebraic methods.

Try solving on your own before revealing the answer!

Q3. Solve: \( \frac{1}{x-5} - \frac{3}{x+7} = \frac{12}{x^2 + 2x - 35} \)

Background

Topic: Solving Rational Equations with Quadratic Denominators

This question tests your ability to solve rational equations where the denominators involve quadratic expressions.

Key Terms and Formulas:

Quadratic denominator: A denominator that is a quadratic expression.
Factor quadratic expressions to find common denominators.

Step-by-Step Guidance

Factor the quadratic denominator \( x^2 + 2x - 35 \) to see if it can be written as a product of two binomials.
Express all denominators in terms of their factors to identify the LCD.
Multiply both sides of the equation by the LCD to eliminate denominators.
Simplify and solve the resulting equation for \( x \).

Try solving on your own before revealing the answer!

Q4. Rewrite each expression as a radical in simplest form. Show work and simplify all values.

a) \( \sqrt{32} \)
b) \( 4\sqrt{-72} \)
c) \( 25^{1/2} \)
d) \( 27^{2/3} \)
e) \( 64^{-2/3} \)

Background

Topic: Simplifying Radicals and Rational Exponents

This question tests your ability to convert between radical and exponential forms and to simplify expressions involving roots and powers.

Key Terms and Formulas:

\( a^{m/n} = \sqrt[n]{a^m} \)
\( \sqrt{a} \) is the principal square root of \( a \).
Negative exponents indicate reciprocals: \( a^{-n} = \frac{1}{a^n} \).

Step-by-Step Guidance

For each part, rewrite the expression using radical notation if it is in exponential form, or vice versa.
Simplify the radical or exponent by factoring out perfect squares, cubes, etc., as appropriate.
For negative exponents, rewrite as a reciprocal.
For imaginary numbers (like \( \sqrt{-72} \)), express in terms of \( i \).

Try simplifying each expression before checking the answer!

Q5. Factor Completely: \( x^3 + 2x^2 - 9x - 18 \)

Background

Topic: Factoring Polynomials

This question tests your ability to factor cubic polynomials completely.

Key Terms and Formulas:

Factoring by grouping: Group terms to factor common factors.
Factor theorem: If \( f(a) = 0 \), then \( (x - a) \) is a factor.

Step-by-Step Guidance

Group the terms in pairs: \( (x^3 + 2x^2) + (-9x - 18) \).
Factor out the greatest common factor from each group.
Look for a common binomial factor and factor it out.
Check if the remaining factor can be factored further.

Try factoring before revealing the answer!

Q6. Factor Completely: \( 48y^4 - 3y^2 \)

Background

Topic: Factoring Polynomials

This question tests your ability to factor out the greatest common factor and recognize patterns.

Key Terms and Formulas:

Greatest common factor (GCF): The largest factor shared by all terms.
Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \).

Step-by-Step Guidance

Factor out the GCF from both terms.
Check if the remaining expression can be factored further (e.g., as a difference of squares).

Try factoring before revealing the answer!

Q7. Solve the equation by factoring: \( x^2 - 15x + 36 = 0 \)

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to factor quadratic equations and use the zero product property.

Key Terms and Formulas:

Quadratic equation: \( ax^2 + bx + c = 0 \).
Zero product property: If \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).

Step-by-Step Guidance

Factor the quadratic expression on the left side.
Set each factor equal to zero.
Solve each resulting equation for \( x \).

Try factoring and solving before revealing the answer!

Q8. Solve the equation by factoring: \( 2x^2 + 9x = 35 \)

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to rearrange and factor quadratic equations.

Key Terms and Formulas:

Quadratic equation: \( ax^2 + bx + c = 0 \).
Zero product property.

Step-by-Step Guidance

Move all terms to one side to set the equation to zero.
Factor the quadratic expression.
Set each factor equal to zero and solve for \( x \).

Try solving before revealing the answer!

Q9. Solve the equation by the square root method and write the solution in exact form: \( (x - 7)^2 = 20 \)

Background

Topic: Solving Quadratic Equations by Square Roots

This question tests your ability to use the square root property to solve quadratic equations.

Key Terms and Formulas:

Square root property: If \( x^2 = a \), then \( x = \pm \sqrt{a} \).

Step-by-Step Guidance

Take the square root of both sides of the equation, remembering to include both the positive and negative roots.
Isolate \( x \) by adding 7 to both sides.

Try solving before revealing the answer!

Q10. Solve the equation by completing the square and write the solution in exact form: \( x^2 + 6x - 3 = 0 \)

Background

Topic: Completing the Square

This question tests your ability to solve quadratic equations by completing the square.

Key Terms and Formulas:

Completing the square: Transforming \( x^2 + bx \) into \( (x + d)^2 \) by adding and subtracting \( (b/2)^2 \).

Step-by-Step Guidance

Move the constant term to the other side of the equation.
Add \( (b/2)^2 \) to both sides to complete the square.
Rewrite the left side as a squared binomial.
Take the square root of both sides and solve for \( x \).

Try completing the square before revealing the answer!

Q11. Solve the equation by using the quadratic formula and write the solution in exact form: \( 2x^2 = 4x - 3 \)

Background

Topic: Quadratic Formula

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

Key Terms and Formulas:

Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Step-by-Step Guidance

Rearrange the equation into standard quadratic form \( ax^2 + bx + c = 0 \).
Identify the coefficients \( a \), \( b \), and \( c \).
Plug the coefficients into the quadratic formula.
Simplify under the square root (the discriminant).

Try applying the quadratic formula before revealing the answer!

Q12. A machine produces open boxes using square sheets of metal. The machine cuts 5-inch by 5-inch squares from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 2000 cubic inches, find the length and width of the open box.

Background

Topic: Applications of Quadratic Equations (Volume)

This question tests your ability to set up and solve equations based on geometric applications, specifically volume of a box.

Key Terms and Formulas:

Volume of a box: \( V = l \times w \times h \)
Dimensions after cutting: If \( x \) is the original side, after cutting 5 inches from each corner, the new length and width are \( x - 2 \times 5 \).

Step-by-Step Guidance

Let \( x \) be the original length of the square sheet.
After cutting, the length and width become \( x - 2 \times 5 \).
The height of the box is 5 inches.
Set up the equation for volume: \( (x - 10)^2 \times 5 = 2000 \).
Solve for \( x \) and then find the length and width.

Try setting up and solving the equation before revealing the answer!

Q13. Solve the equation by factoring: \( 5x^4 - 45x^2 = 0 \)

Background

Topic: Factoring Polynomials

This question tests your ability to factor out common terms and solve higher-degree equations.

Key Terms and Formulas:

Factor out the greatest common factor.
Set each factor equal to zero and solve for \( x \).

Step-by-Step Guidance

Factor out the GCF from both terms.
Recognize the remaining expression as a difference of squares and factor further.
Set each factor equal to zero and solve for \( x \).

Try factoring and solving before revealing the answer!

Q14. Solve the equation by factoring: \( 6x^4 - 6x^3 - 72x^2 = 0 \)

Background

Topic: Factoring Polynomials

This question tests your ability to factor out the GCF and factor higher-degree polynomials.

Key Terms and Formulas:

Factor out the GCF.
Factor the remaining quadratic or cubic expression.

Step-by-Step Guidance

Factor out the GCF from all terms.
Factor the remaining polynomial if possible.
Set each factor equal to zero and solve for \( x \).

Try factoring and solving before revealing the answer!

Q15. Solve: \( 1 - 2x = x + 7 \)

Background

Topic: Solving Linear Equations

This question tests your ability to solve basic linear equations for a single variable.

Key Terms and Formulas:

Linear equation: An equation of the form \( ax + b = c \).

Step-by-Step Guidance

Move all terms involving \( x \) to one side and constants to the other.
Combine like terms.
Isolate \( x \) by dividing both sides by the coefficient of \( x \).

Try solving before revealing the answer!

Q16. Solve the equation: \( (x + 5)^{2/3} = 4 \)

Background

Topic: Solving Equations with Rational Exponents

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

Key Terms and Formulas:

\( a^{m/n} = \sqrt[n]{a^m} \)
To solve, raise both sides to the reciprocal power.

Step-by-Step Guidance

Raise both sides of the equation to the reciprocal of \( 2/3 \), which is \( 3/2 \).
Simplify the right side.
Solve for \( x \) by isolating it.

Try solving before revealing the answer!

Q17. Solve: \( |5x + 2| + 4 = 13 \)

Background

Topic: Solving Absolute Value Equations

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

Key Terms and Formulas:

Absolute value: \( |a| = a \) if \( a \geq 0 \), \( -a \) if \( a < 0 \).
To solve, isolate the absolute value and set up two equations.

Step-by-Step Guidance

Subtract 4 from both sides to isolate the absolute value.
Set up two equations: one with the expression inside the absolute value equal to the positive value, and one equal to the negative value.
Solve each equation for \( x \).

Try solving before revealing the answer!

Q18. a) Solve and express the solution using interval notation. b) Graph the solution set on the number line. \( \frac{x}{6} - \frac{1}{6} \leq \frac{x}{3} + 2 \)

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve inequalities and express solutions in interval notation and graphically.

Key Terms and Formulas:

Linear inequality: An inequality involving a linear expression.
Interval notation: A way to express solution sets using intervals.

Step-by-Step Guidance

Combine like terms and move all terms involving \( x \) to one side.
Isolate \( x \) and solve the inequality.
Express the solution in interval notation.
Sketch the solution on a number line.

Try solving and graphing before revealing the answer!

Q19. a) Solve and express the solution using interval notation. b) Graph the solution set on the number line. \( \frac{x - 2}{3} \geq \frac{x - 4}{4} + \frac{7}{12} \)

Background

Topic: Solving Linear Inequalities with Fractions

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

Key Terms and Formulas:

Find a common denominator to combine terms.
Isolate \( x \) and solve the inequality.

Step-by-Step Guidance

Find the least common denominator for all terms.
Multiply both sides by the LCD to clear denominators.
Simplify and solve for \( x \).
Express the solution in interval notation and graph it.

Try solving and graphing before revealing the answer!

Q20. a) Solve and express the solution using interval notation. b) Graph the solution set on the number line. \( |x + 1| + 3 \leq 9 \)

Background

Topic: Solving Absolute Value Inequalities

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

Key Terms and Formulas:

Absolute value inequality: \( |a| \leq b \) means \( -b \leq a \leq b \).

Step-by-Step Guidance

Subtract 3 from both sides to isolate the absolute value.
Set up the compound inequality based on the absolute value property.
Solve for \( x \) and express the solution in interval notation.
Graph the solution on a number line.

Try solving and graphing before revealing the answer!

Q21. a) Solve and express the solution using interval notation. b) Graph the solution set on the number line. \( -2|x - 4| < -4 \)

Background

Topic: Solving Absolute Value Inequalities

This question tests your ability to manipulate and solve inequalities involving absolute value expressions.

Key Terms and Formulas:

Isolate the absolute value before setting up the compound inequality.

Step-by-Step Guidance

Divide both sides by -2, remembering to reverse the inequality sign.
Set up the compound inequality for the absolute value.
Solve for \( x \) and express the solution in interval notation.
Graph the solution on a number line.