Question 30·Hard·Linear Inequalities in One or Two Variables
A transportation researcher surveyed 400 commuters about their daily travel time. Of the first 250 respondents, 68 reported that their one-way commute takes more than one hour. Among the remaining 150 respondents, reported a commute of more than one hour. If less than 25% of all 400 commuters surveyed reported a commute of more than one hour, which of the following inequalities best describes the possible values of ?
For inequality word problems with percentages, first define the variable clearly and write an expression for the total quantity in question (here, total commuters with long commutes). Next, convert the percentage condition into a numerical expression, and set up the inequality with the correct direction symbol based on phrases like "less than" or "at most." Finally, include any natural restrictions on the variable (like a maximum number of people) and choose the option that matches both the inequality and the domain.
Hints
Express the total number with long commutes
How can you write the total number of commuters with more than a one-hour commute using the 68 from the first group and the unknown from the second group?
Turn the percentage into a number to compare
You are told that less than 25% of 400 commuters have long commutes. How can you write 25% of 400 as a multiplication expression?
Set up the inequality
Place your expression for the total number with long commutes on one side and your expression for 25% of 400 on the other side. Which inequality symbol should you use to represent "less than"?
Remember the limit on
counts how many of the remaining 150 commuters have long commutes. What inequality must satisfy because there are only 150 people left?
Desmos Guide
Graph the total long-commute count and the 25% threshold
In Desmos, use in place of . Enter two expressions on separate lines: y = x + 68 (the total number of long-commute commuters) and y = 0.25 * 400 (the 25% of 400 threshold). You will see a slanted line and a horizontal line.
Use the graph to understand valid -values
Focus on (since cannot be more than 150 commuters). Look where the line lies below the horizontal line . The -values in this region show which values of make the "less than 25%" condition true; compare this to the answer choices’ inequalities.
Step-by-step Explanation
Identify what represents and total commuters with long commutes
From the problem:
- Among the first 250 respondents, 68 have commutes longer than one hour.
- Among the remaining 150 respondents, have commutes longer than one hour.
So the total number of commuters with more than a one-hour commute is:
Translate "less than 25% of all 400 commuters" into a number
There are 400 commuters in total. The problem says less than 25% of them reported a commute of more than one hour.
Compute 25% of 400:
We don't actually have to simplify this product for the inequality, since the answer choices all keep it as .
Write the inequality comparing totals
"Less than 25%" means the number of long-commute commuters must be less than 25% of 400.
That gives the inequality:
This compares the total number with long commutes on the left to 25% of all commuters on the right.
Include the restriction on and match the answer choice
Because is the number of people (among the remaining 150) with commutes over an hour, cannot be more than 150, so .
Putting everything together, the inequality that describes the possible values of is:
This matches answer choice C.