College

Which situation can be modeled by the inequality [tex]3x + 5 \geq 12[/tex]?

A. Ellie's car holds at least 12 gallons of gasoline. She needs 5 gallons of gasoline to get to work each week. How many gallons of gasoline, [tex]x[/tex], will be in her car at the end of the work week?

B. April writes in her journal for 5 minutes each day. She increases her writing time by 3 minutes each week. April wants to write no more than 12 minutes each day. For how many weeks, [tex]x[/tex], should April increase her writing time?

C. Ivy wants to bike at least 12 miles every day. She begins by biking 3 miles. She adds 5 miles to her ride every month. How many months, [tex]x[/tex], will it take Ivy to reach her goal?

D. Amir has written 5 thank-you notes for presents. He has at least 12 notes to write. Amir wants to finish the notes in the next 3 days. How many notes, [tex]x[/tex], will Amir need to write each day?

Answer :

To determine which situation can be modeled by the inequality [tex]\(3x + 5 \geq 12\)[/tex], let's break down the inequality and see which scenario fits.

First, let's simplify the inequality:

1. Start with the inequality: [tex]\(3x + 5 \geq 12\)[/tex].
2. Subtract 5 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\(3x \geq 7\)[/tex].
3. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\(x \geq \frac{7}{3}\)[/tex].

This means that [tex]\(x\)[/tex] (the quantity we're interested in for the situation) must be at least 2.33 (approximately). Now, let's match this with the situations:

1. Ellie's car situation:
She uses 5 gallons to get to work, and the car holds at least 12 gallons. This doesn't seem to involve a step-wise increase (like adding a certain amount regularly).

2. April's writing situation:
April starts writing for 5 minutes and adds 3 minutes weekly to reach no more than 12 minutes. This situation is about decreasing to not exceed a certain limit, not about reaching at least a certain amount.

3. Ivy's biking situation:
Ivy starts biking 3 miles and adds 5 miles each month, aiming to bike at least 12 miles. This matches our inequality because she adds 5 miles monthly to meet or exceed the goal of 12 miles. We begin with 3 miles and need to reach at least 12 miles, which fits the structure of adding a set amount regularly.

4. Amir's notes situation:
Amir has written 5 notes and needs to complete at least 12 in total; the time frame for writing notes each day is 3 days. This situation focuses on distributing already known work (notes) and time rather than regularly accumulating towards a target.

The situation that best matches the inequality [tex]\(3x + 5 \geq 12\)[/tex] is Ivy's biking situation, where the monthly increase adds towards a minimum biking goal.