Question

1. multi-step jeds online music club allows him to download 25 songs per month for $14.99. additional songs cost $1.29 each. a. write an inequality to represent this situation. let t be his monthly spending limit and m represent the total number of songs downloaded. a 1.29m ≤ t + 10.01 b 1.29m ≤ t + 17.26 c 1.29t ≤ m + 10.01 d 1.29t ≤ m + 17.26 b. which of the following statements best describes the number of songs jed can download each month without going over his spending limit? a jed will not go over his spending limit of $30 if he downloads 37 songs. b jed will not go over his spending limit of $40 if he downloads 45 songs. c jed can download a maximum of 36 songs if his budget is $30. d jed can download a maximum of 36 songs if his budget is $40. c. suppose that the music club changes its plan so that 50 songs can be downloaded a month. how will the inequality representing this inequality change? a the coefficient of t will decrease. b the coefficient of t will increase. c the coefficient of m will increase. d the constant will increase.

Explanation:

Part a

Step1: Analyze the cost structure

The base cost for 25 songs is $14.99. If \(m\) is the total number of songs, the number of additional songs is \(m - 25\). The cost of additional songs is \(1.29(m - 25)=1.29m-1.29\times25 = 1.29m - 32.25\). The total cost \(C\) is \(C=1.29m-32.25\) (since the first 25 songs are already accounted for in the base - like structure). We want \(C\leq t\) (where \(t\) is the spending limit). Rearranging gives \(1.29m\leq t + 32.25-14.99\). Calculate \(32.25 - 14.99=17.26\). So the inequality is \(1.29m\leq t + 17.26\).

Part b

Step1: Check option A

If \(t = 30\) and \(m = 37\), then the cost \(C=14.99+1.29\times(37 - 25)=14.99+1.29\times12=14.99 + 15.48=30.47>30\).

Step2: Check option B

If \(t = 40\) and \(m = 45\), then the cost \(C=14.99+1.29\times(45 - 25)=14.99+1.29\times20=14.99 + 25.8=40.79>40\).

Step3: Check option C

If \(t = 30\), then \(1.29m\leq30 + 17.26\). So \(1.29m\leq47.26\), and \(m\leq\frac{47.26}{1.29}\approx36.64\). Since \(m\) is an integer, \(m = 36\). The cost for \(m = 36\) is \(14.99+1.29\times(36 - 25)=14.99+1.29\times11=14.99+14.19 = 29.18\leq30\).

Step4: Check option D

If \(t = 40\), then \(1.29m\leq40 + 17.26\). So \(1.29m\leq57.26\), and \(m\leq\frac{57.26}{1.29}\approx44.4\). Since \(m\) is an integer, \(m = 44
eq36\).

Part c

Step1: Analyze the original and new cost structure

Original inequality: \(C = 1.29m-1.29\times25\leq t\). New inequality: \(C'=1.29m-1.29\times50\leq t\). The constant term in the inequality \(1.29m\leq t+1.29\times k\) (where \(k\) is the number of included songs) changes. Since \(1.29\times50>1.29\times25\), the constant term increases.

Answer:

a. B. \(1.29m\leq t + 17.26\)
b. C. Jed can download a maximum of 36 songs if his budget is $30.
c. D. The constant will increase.