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Chapter 5: Problem 72

Let \(x\) represent the number of country songs that Sierra puts on a playlist on her portable media player. Let \(y\) represent the number of rock songs that she puts on the playlist. For parts (a)-(e), write an inequality to represent the given statement. a. Sierra will put at least 6 country songs on the playlist. b. Sierra will put no more than 10 rock songs on the playlist. c. Sierra wants to limit the length of the playlist to at most 20 songs. d. The number of country songs cannot be negative. e. The number of rock songs cannot be negative. f. Graph the solution set to the system of inequalities from parts (a)-(e).

Short Answer

Expert verified
a. \( x \geq 6 \) b. \( y \leq 10 \) c. \( x + y \leq 20 \) d. \( x \geq 0 \) e. \( y \geq 0 \)

Step by step solution

01

Represent 'at least 6 country songs'

Define the inequality for the number of country songs. Since 'at least 6' means 6 or more, the inequality is: \[ x \geq 6 \]
02

Represent 'no more than 10 rock songs'

Define the inequality for the number of rock songs. Since 'no more than 10' means 10 or fewer, the inequality is: \[ y \leq 10 \]
03

Limit the playlist to at most 20 songs

Sum the total number of songs and ensure they do not exceed 20. This gives the inequality: \[ x + y \leq 20 \]
04

Country songs cannot be negative

Express that the number of country songs must be non-negative with the inequality: \[ x \geq 0 \]
05

Rock songs cannot be negative

Express that the number of rock songs must be non-negative with the inequality: \[ y \geq 0 \]
06

Graph the inequalities

Graph each of the inequalities on a coordinate plane: - For \( x \geq 6 \), draw a vertical line at \( x = 6 \) and shade to the right. - For \( y \leq 10 \), draw a horizontal line at \( y = 10 \) and shade below. - For \( x + y \leq 20 \), draw a line for \( x + y = 20 \) and shade below the line. - For \( x \geq 0 \), shade to the right of the y-axis. - For \( y \geq 0 \), shade above the x-axis.The solution region is where all shaded areas overlap.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Systems of Inequalities
A system of inequalities consists of multiple inequalities considered together. In this exercise, we have several conditions to meet, like the number of country and rock songs Sierra wants in her playlist. Each of these conditions is an inequality:
  • The number of country songs must be at least 6: \( x \geq 6 \)
  • The number of rock songs must be at most 10: \( y \leq 10 \)
  • The total number of songs cannot exceed 20: \( x + y \leq 20 \)
  • Both numbers of songs must be non-negative: \( x \geq 0 \) and \( y \geq 0 \)
    • These inequalities together form a system that we need to solve simultaneously.
    The solution to this system of inequalities is the set of all possible values of \( x \) and \( y \) that satisfy all these conditions at the same time.
    By understanding each inequality individually first, we can then combine them to find the overall solution.
Graphing Inequalities
Graphing inequalities involves plotting each inequality on a coordinate plane. This helps visually represent all possible solutions for \( x \) and \( y \). To graph the inequalities from our exercise:
  • For \( x \geq 6 \), draw a vertical line at \( x = 6 \) and shade the region to the right because \( x \) should be 6 or more.
  • For \( y \leq 10 \), draw a horizontal line at \( y = 10 \) and shade the region below the line because \( y \) should be 10 or fewer.
  • For \( x + y \leq 20 \), plot the line \( x + y = 20 \). This line slopes downwards from (20,0) to (0,20). Shade below this line where the total number of songs is at most 20.
  • For \( x \geq 0 \) and \( y \geq 0 \), simply shade the first quadrant since song counts can't be negative.
The solution to the system will be the region where all these shaded areas overlap on the graph. This visual solution makes it easier to understand what combinations of \( x \) and \( y \) satisfy all the conditions simultaneously.
Algebraic Expressions
Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. In this exercise, we use algebraic expressions to set limits on the numbers of songs:
  • The expression \( x \geq 6 \) states that \( x \), the number of country songs, has to be at least 6.
  • The expression \( y \leq 10 \) asserts that \( y \), the number of rock songs, can be at most 10.
  • The expression \( x + y \leq 20 \) ensures the total number of songs (country plus rock) doesn't exceed 20.
These expressions help us set up clear rules and conditions that our solutions need to meet. By working with algebraic expressions, we can translate word problems into solvable mathematical statements.
Real-World Applications of Algebra
Algebra isn't just about solving equations on paper; it has real-world applications that we encounter daily. Take this exercise, for example:
  • Sierra is organizing her playlist, and she has specific requirements. Using algebra, she can ensure her playlist meets all these requirements efficiently.
  • In other contexts, such as budgeting or planning events, algebraic inequalities allow us to set limits and optimize resources.
  • Imagine industries where resources are allocated under constraints; systems of inequalities help in optimizing usage and planning.
  • From planning our daily schedules to managing large-scale projects, algebra helps by giving us tools to create and solve conditions systematically.
Through this exercise, we see how a seemingly simple problem of arranging songs can involve detailed algebraic thinking and lead to practical, optimal solutions.

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Most popular questions from this chapter

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} x^{2}+y^{2}=40 \\ y=-x^{2}+8.5 \end{array} $$

Josh makes \(\$ 24 /\) hr tutoring chemistry and \(\$ 20 / \mathrm{hr}\) tutoring math. Let \(x\) represent the number of hours per week he spends tutoring chemistry. Let \(y\) represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring \(x\) hours of chemistry and \(y\) hours of math. b. The time that Josh devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. \- The number of hours spent tutoring each subject cannot be negative. \- Due to the academic demands of his own classes he tutors at most \(18 \mathrm{hr}\) per week. \- The tutoring center requires that he tutors math at least 4 hr per week. \- The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many hours tutoring math and how many hours tutoring chemistry should Josh work to maximize his income? g. What is the maximum income? h. Explain why Josh's maximum income is found at a point on the line \(x+y=18\).

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