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All courses Math Foundations · MF.EQ.12 39 of 60
Interpret algebraic solutions and state them as meaningful contextual answers, including restrictions or whole-number constraints when needed.

Turn a solved inequality into a context range

Problem
The solved inequality is x ≤ 5, where x is a nonnegative whole-number count of extra guests. Give the contextual set.
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Hint

Start by translating x <= 5 into words: x can be 5 or fewer. Then remember x is counting extra guests.

A solved inequality gives a whole set of possible values, not just one value. Since x counts guests, use only whole numbers that are not negative.

Solution walkthrough

01

Identify what the variable means

\[x~=~\text{number}~\text{of}~\text{extra}~\text{guests}\]

The problem already tells us that x counts extra guests, so x must represent a whole number of people.

02

Read the inequality correctly

\[x~\le~5\]

The symbol <= means "less than or equal to," so the number of extra guests can be 5 or any smaller value.

03

Use the guest-count restrictions

\[0~\le~x~\le~5,~\text{and}~x~\text{is}~a~\text{whole}~\text{number}\]

A number of guests cannot be negative, and you cannot have part of a guest. So only whole numbers from 0 through 5 are allowed.

04

State the possible values in context

\[x~∈~{0,~1,~2,~3,~4,~5}~\text{extra}~\text{guests}\]

This means there can be at most 5 extra guests. Any number of extra guests from 0 to 5 works.

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Another way

  1. Check the boundary: if x = 5, then host + 12 friends + 5 extra guests = 18, so 5 works. If x = 6, then the total would be 19, which is too many.

!

Common mistake

Saying the answer is only 5 extra guests. The inequality x <= 5 means 5 is the maximum, not the only value. It also allows 0, 1, 2, 3, and 4 extra guests.