Textbook Question
In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (- ∞, 5.5)
881
views
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 12
An electronic pass for a toll road costs \$30. The toll is normally \$5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the total cost without the pass is the same as the total cost with the pass.
Verified step by step guidance1
Define the variable: let \(x\) represent the number of times the road is used.
Write the expression for the total cost without the electronic pass: since each toll costs \(5.00\), the total cost is \$5x$.
Write the expression for the total cost with the electronic pass: the pass costs \(30\) upfront, and each toll is reduced by 30%, so each toll costs \(5 \times (1 - 0.30) = 5 \times 0.70 = 3.5\). Therefore, the total cost is \(30 + 3.5x\).
Set the total costs equal to find the break-even point: \(5x = 30 + 3.5x\).
Solve the equation for \(x\) by isolating \(x\) on one side: subtract \$3.5x\( from both sides to get \(5x - 3.5x = 30\), then simplify and solve for \)x$.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Setting Up Equations for Cost Comparison
To solve the problem, you need to express the total costs with and without the electronic pass as algebraic expressions. This involves defining variables (e.g., number of uses) and writing equations that represent each total cost scenario for comparison.
Recommended video:
05:56
Introduction to Rational Equations
Percentage Discount Calculation
Understanding how to calculate a percentage discount is essential. Here, a 30% reduction on the $5 toll means multiplying $5 by 0.30 to find the discount amount, then subtracting it from the original toll to find the reduced toll cost.
Recommended video:
5:37Introduction to Probability
Solving Linear Equations
Once the cost expressions are set equal, solving for the variable requires knowledge of linear equations. This involves isolating the variable on one side through algebraic manipulation to find the number of uses where costs are equal.
Recommended video:
04:02Solving Linear Equations with Fractions
Related Practice
Textbook Question
Plot the given point in a rectangular coordinate system. (- 5/2, 3/2)
1221
views
Textbook Question
Plot the given point in a rectangular coordinate system. (7/2, - 3/2)
861
views
Textbook Question
Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(x + 3) = x - 3
403
views
Textbook Question
In Exercises 9–20, find each product and write the result in standard form. (- 5 + 4i)(3 + i)
841
views
Textbook Question
In Exercises 1–26, solve and check each linear equation. 7x + 4 = x + 16
696
views