Skip to main content
Back

College Algebra Exam 1 Review – Step-by-Step Guidance

NotesPracticeVideo lessons

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Q1. Use your graphing utility to approximate the intercepts rounded to three decimal places for the equation .

Background

Topic: Polynomial Functions and Graphs

This question tests your ability to find the x- and y-intercepts of a cubic polynomial using a graphing calculator or software.

Key Terms and Formulas:

  • x-intercept: The value(s) of where .

  • y-intercept: The value of when .

Step-by-Step Guidance

  1. To find the y-intercept, substitute into the equation: .

  2. To find the x-intercepts, set and solve .

  3. Factor the equation if possible to make solving easier.

  4. Use your graphing utility to plot the function and identify the points where the graph crosses the x-axis (these are the x-intercepts).

Try solving on your own before revealing the answer!

Q2. Solve the equation graphically: . Round your solution(s) to two decimal places.

Background

Topic: Solving Equations Graphically

This question asks you to use a graphing utility to find the approximate solutions to a quartic equation.

Key Terms and Formulas:

  • Graphical Solution: Plot both sides of the equation and find where they intersect.

  • Quartic Equation: An equation involving .

Step-by-Step Guidance

  1. Rewrite the equation so all terms are on one side: .

  2. Simplify the equation: Combine like terms.

  3. Use your graphing utility to plot the simplified equation.

  4. Identify the x-values where the graph crosses the x-axis; these are your approximate solutions.

Try solving on your own before revealing the answer!

Q3a. Solve the equation algebraically: .

Background

Topic: Solving Polynomial Equations

This question tests your ability to combine like terms and solve for in a quadratic equation.

Key Terms and Formulas:

  • Quadratic Equation: An equation involving .

  • Combining Like Terms: Add or subtract terms with the same variable and exponent.

Step-by-Step Guidance

  1. Expand and simplify each term: , , .

  2. Combine all terms on one side and constants on the other.

  3. Move all terms to one side to set the equation equal to zero.

  4. Factor or use the quadratic formula to solve for .

Try solving on your own before revealing the answer!

Q4. A car dealer reduces the list price of its vehicles by 14%. If one vehicle has a sale price of $28,809.99, what was its original list price to the nearest cent?

Background

Topic: Percentages and Reverse Calculations

This question tests your ability to work backwards from a sale price to find the original price before a percentage reduction.

Key Terms and Formulas:

  • Percent Decrease: The sale price is a certain percent less than the original price.

  • Formula:

Step-by-Step Guidance

  1. Let be the original price. The sale price is .

  2. Set up the equation: .

  3. Solve for by dividing both sides by .

Try solving on your own before revealing the answer!

Q5a. If a rock falls from a height of 20 meters, the height in meters after seconds is given by . What is the height of the rock after 1.5 seconds?

Background

Topic: Quadratic Functions and Applications

This question tests your ability to substitute values into a quadratic function modeling free fall.

Key Terms and Formulas:

  • Free Fall Equation:

  • Substitution: Plug in .

Step-by-Step Guidance

  1. Write the equation: .

  2. Substitute into the equation.

  3. Calculate .

  4. Subtract this value from 20 to find the height.

Try solving on your own before revealing the answer!

Q5b. When does the rock strike the ground? (approximate to two decimal places)

Background

Topic: Quadratic Equations and Applications

This question tests your ability to solve for the time when the height is zero.

Key Terms and Formulas:

  • Free Fall Equation:

  • Ground Strike: Set and solve for .

Step-by-Step Guidance

  1. Set : .

  2. Rearrange to solve for : .

  3. Divide both sides by to isolate .

  4. Take the square root of both sides to solve for .

Try solving on your own before revealing the answer!

Q6a. Solve for : .

Background

Topic: Solving Cubic Equations

This question tests your ability to solve a cubic equation algebraically.

Key Terms and Formulas:

  • Cubic Equation: An equation involving .

  • Algebraic Solution: Move all terms to one side and factor or use other methods.

Step-by-Step Guidance

  1. Move all terms to one side: .

  2. Try factoring or use rational root theorem to find possible solutions.

  3. Test possible integer or rational roots.

  4. If necessary, use synthetic division or other methods to factor further.

Try solving on your own before revealing the answer!

Q7a. Solve the equation algebraically: .

Background

Topic: Linear Equations

This question tests your ability to solve a simple linear equation for .

Key Terms and Formulas:

  • Linear Equation: An equation of the form .

Step-by-Step Guidance

  1. Add 3 to both sides: .

  2. Divide both sides by 4 to solve for .

Try solving on your own before revealing the answer!

Q8a. Solve the inequalities: and ; give your answers in interval notation.

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas:

  • Linear Inequality: An inequality involving .

  • Interval Notation: Expressing the solution as a range of values.

Step-by-Step Guidance

  1. Solve for .

  2. Solve .

  3. Combine the two solutions to find the intersection.

  4. Express the final answer in interval notation.

Try solving on your own before revealing the answer!

Q9a. Solve the system of equations:

Background

Topic: Systems of Linear Equations

This question tests your ability to solve a system of two linear equations using substitution or elimination.

Key Terms and Formulas:

  • System of Equations: Two or more equations to be solved together.

  • Substitution/Elimination: Methods for solving systems.

Step-by-Step Guidance

  1. Solve one equation for one variable (e.g., in the second equation).

  2. Substitute this expression into the other equation.

  3. Solve for the remaining variable.

  4. Back-substitute to find the other variable.

Try solving on your own before revealing the answer!

Q10. If the area of a rectangle is 108 square inches and the perimeter is 42 inches, what are the dimensions of the rectangle?

Background

Topic: Quadratic Equations and Geometry

This question tests your ability to set up and solve a system involving area and perimeter formulas.

Key Terms and Formulas:

  • Area:

  • Perimeter:

Step-by-Step Guidance

  1. Let = length and = width.

  2. Set up the equations: and .

  3. Solve one equation for one variable and substitute into the other.

  4. Solve the resulting quadratic equation for or .

Try solving on your own before revealing the answer!

Q11. A motorboat travels on a river that has a current of 3 mph. The trip upstream takes 5 hours and the return trip downstream takes 2.5 hours. What is the speed of the boat?

Background

Topic: Linear Equations and Motion Problems

This question tests your ability to set up and solve equations involving rates, times, and distances.

Key Terms and Formulas:

  • Distance Formula:

  • Upstream Speed:

  • Downstream Speed:

Step-by-Step Guidance

  1. Let be the speed of the boat in still water.

  2. Set up equations for upstream and downstream trips using .

  3. Since the distance is the same both ways, set the two expressions equal.

  4. Solve for .

Try solving on your own before revealing the answer!

Q12. Donna invested $33,000 and received a total of $970 in interest after one year. If part of the money returned 4% interest and the rest returned 2.25% interest, how much did she invest at each rate?

Background

Topic: Systems of Equations and Investment Problems

This question tests your ability to set up and solve a system of equations based on investment returns.

Key Terms and Formulas:

  • Interest Formula:

  • System of Equations: Total invested and total interest equations.

Step-by-Step Guidance

  1. Let be the amount invested at 4%, at 2.25%.

  2. Set up equations: and .

  3. Solve one equation for one variable and substitute into the other.

  4. Solve for and .

Try solving on your own before revealing the answer!

Q13. How much water should be added to 300 milliliters of a 60% acid solution to make a 50% acid solution?

Background

Topic: Mixture Problems and Linear Equations

This question tests your ability to set up and solve a mixture problem using percentages.

Key Terms and Formulas:

  • Mixture Formula:

Step-by-Step Guidance

  1. Let be the amount of water to add.

  2. Set up the equation: .

  3. Solve for .

Try solving on your own before revealing the answer!

Q14. An 8% solution and a 15% solution are to be mixed to obtain 20 ounces of a 10% solution. How much of each should be used? Round answers to 2 decimal places.

Background

Topic: Mixture Problems and Systems of Equations

This question tests your ability to set up and solve a system for mixture concentrations.

Key Terms and Formulas:

  • Mixture Formula:

  • Total Volume:

Step-by-Step Guidance

  1. Let be ounces of 8% solution, of 15% solution.

  2. Set up the two equations: and .

  3. Solve one equation for one variable and substitute into the other.

  4. Solve for and .

Try solving on your own before revealing the answer!

Q15. Two cyclists leave a city at the same time, one going east and the other going west. The westbound cyclist bikes 5 mph faster than the eastbound cyclist. After 6 hours they are 246 miles apart. How fast is each cyclist riding?

Background

Topic: Linear Equations and Motion Problems

This question tests your ability to set up and solve equations involving rates, times, and distances.

Key Terms and Formulas:

  • Distance Formula:

Step-by-Step Guidance

  1. Let be the speed of the eastbound cyclist; westbound is .

  2. Each travels for 6 hours: and .

  3. Total distance apart: .

  4. Solve for .

Try solving on your own before revealing the answer!

Q16c. Find an equation of the line parallel to and through the point (4, 1). Give the answer in slope-intercept form if possible.

Background

Topic: Linear Equations and Slope

This question tests your ability to find the equation of a line parallel to a given line and passing through a specific point.

Key Terms and Formulas:

  • Slope-Intercept Form:

  • Parallel Lines: Same slope as the given line.

Step-by-Step Guidance

  1. Find the slope of the given line by rewriting in slope-intercept form.

  2. Use the same slope for your new line.

  3. Plug the point (4, 1) into the equation to solve for .

  4. Write the final equation in form.

Try solving on your own before revealing the answer!

Q17. Find the intercepts of the line . Use the intercepts to graph the line.

Background

Topic: Linear Equations and Graphing

This question tests your ability to find x- and y-intercepts and use them to graph a line.

Key Terms and Formulas:

  • x-intercept: Set and solve for .

  • y-intercept: Set and solve for .

Step-by-Step Guidance

  1. Set in and solve for .

  2. Set and solve for .

  3. Plot these points on a graph and draw the line.

Try solving on your own before revealing the answer!

Q18. An electric company charges a flat monthly fee of Cx$ kilowatt-hours in a month.

Background

Topic: Linear Functions and Applications

This question tests your ability to write a linear equation modeling a real-world situation.

Key Terms and Formulas:

  • Linear Equation:

Step-by-Step Guidance

  1. Identify the fixed fee () and the variable rate ().

  2. Write the equation: .

Try solving on your own before revealing the answer!

Q19. The volume of a gas in a closed container varies inversely with its pressure. If the volume is 600 cm³ when the pressure is 150 mm Hg, find the volume when the pressure is 200 mm Hg.

Background

Topic: Inverse Variation

This question tests your ability to use inverse variation to solve for unknowns.

Key Terms and Formulas:

  • Inverse Variation:

Step-by-Step Guidance

  1. Set up the equation: .

  2. Solve for .

  3. Use to find the volume at .

Try solving on your own before revealing the answer!

Q20. The resistance of a wire varies directly with its length and inversely with the square of the diameter. If a wire 1.2 meters long and 0.5 cm in diameter has a resistance of 140 ohms, find the resistance of a wire made from the same material that is 3 meters long and has a diameter of 0.8 cm.

Background

Topic: Direct and Inverse Variation

This question tests your ability to set up and solve a variation equation.

Key Terms and Formulas:

  • Variation Formula:

Step-by-Step Guidance

  1. Set up the equation for the first wire: .

  2. Solve for .

  3. Use to find the resistance for the second wire: .

Try solving on your own before revealing the answer!

Q21a. Find and simplify for .

Background

Topic: Function Evaluation

This question tests your ability to substitute a value into a function and simplify.

Key Terms and Formulas:

  • Function Evaluation: Substitute with the given value.

Step-by-Step Guidance

  1. Write the function: .

  2. Substitute into the function.

  3. Calculate , , and .

  4. Add the results together.

Try solving on your own before revealing the answer!

Q22a. Find the domain of .

Background

Topic: Domain of Rational Functions

This question tests your ability to find the domain of a function with a denominator.

Key Terms and Formulas:

  • Domain: All real values of for which the function is defined.

  • Rational Function: A function with a variable in the denominator.

Step-by-Step Guidance

  1. Identify values of that make the denominator zero: .

  2. Solve for .

  3. The domain excludes this value.

Try solving on your own before revealing the answer!

Q24a. If a rock falls from a height of 25 meters, the height in meters after seconds is . What is the height after 1.2 seconds?

Background

Topic: Quadratic Functions and Applications

This question tests your ability to substitute values into a quadratic function modeling free fall.

Key Terms and Formulas:

  • Free Fall Equation:

Step-by-Step Guidance

  1. Write the equation: .

  2. Substitute into the equation.

  3. Calculate .

  4. Subtract this value from 25 to find the height.

Try solving on your own before revealing the answer!

Pearson Logo

Study Prep