Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Holder inequality: Σ1615 ΣΕΡ j=1 where p > 1 and Σ m=1 1 1 + 1. P q 1 Cauchy-Schwarz inequality: Σx² j=1 m=1 Minkowski inequality: (+1) (Ex)'s (Eur)' (E) where p > 1. + Problem 14: Duality in LP Spaces Problem Statement: Consider IP (2) spaces where 2 is a measure space and 1

Graph the linear inequality: y1 .

Elena needs to earn at least $450 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays $9 an hour and the other as an intern in a genetics lab for $22.50 per hour. How many hours does Elena need to work at each job to earn at least $450 per week? (a) Let x be the number of hours she works teaching swimming and let y be the number of hours she works as an intern. Write an inequality that would model this situation. (b) Graph the inequality. (c) Find three ordered pairs (x, y) that would be solutions to the inequality. Then, explain what that means for Elena.

Veronica works two part time jobs in order to earn enough money to meet her obligations of at least $280 a week. Her job at the day spa pays $10 an hour and her administrative assistant job on campus pays $17.50 an hour. How many hours does Veronica need to work at each job to earn at least $280? (a) Let x be the number of hours she works at the day spa and let y be the number of hours she works as administrative assistant. Write an inequality that would model this situation. (b) Graph the inequality. (c) Find three ordered pairs (x, y) that would be solutions to the inequality. Then, explain what that means for Veronica

Hugh works two part time jobs. One at a grocery store that pays $10 an hour and the other is babysitting for $13 hour. Between the two jobs, Hugh wants to earn at least $260 a week. How many hours does Hugh need to work at each job to earn at least $260? (a) Let x be the number of hours he works at the grocery store and let y be the number of hours he works babysitting. Write an inequality that would model this situation. (b) Graph the inequality. (c) Find three ordered pairs (x, y) that would be solutions to the inequality. Then, explain what that means for Hugh.

Harrison works two part time jobs. One at a gas station that pays $11 an hour and the other is IT troubleshooting for $16.50 an hour. Between the two jobs, Harrison wants to earn at least $330 a week. How many hours does Harrison need to work at each job to earn at least $330? (a) Let x be the number of hours he works at the gas station and let y be the number of (hours he works troubleshooting. Write an inequality that would model this situation. (b) Graph the inequality. (c) Find three ordered pairs (x, y) that would be solutions to the inequality. Then, explain what that means for Harrison.