Skills Beta

Inequalities in Words

Translate words into inequalities before computing: at least is ≥ , at most and no more than are ≤ , more than is > , fewer than is < .

Practice 3: Analyze Data

Part 1 · Hook

Why this matters

A coach wants to know how often a player makes "at least 3" of 5 shots; a teacher counts students who scored "fewer than 3" errors. One word changes which counts are included. Get the boundary wrong and every number after it is wrong too, which is one of the most common slips on the AP® Statistics exam.

Part 2 · Prerequisite check

Quick check before you start

1. Which variable is discrete?

Number of shots made
Time to run a mile
Height
Weight

Show the answer

A count takes separate whole-number values.

Correct: Number of shots made:
Time to run a mile:
Height:
Weight:

2. What proportion of 10 students is 3 students?

0.3
3
0.03
0.7

Show the answer

3/10 = 0.3.

Correct: 0.3:
3:
0.03:
0.7:

Part 3 · See it

See it first

Four number lines for a count from 0 to 8, with the included counts filled in plum. At least 3, X ≥ 3: 3 through 8. More than 3, X > 3: 4 through 8. At most 3, X ≤ 3: 0 through 3. Fewer than 3, X < 3, so X ≤ 2: 0, 1 and 2. — At least 3, more than 3, at most 3 and fewer than 3 on a number line of counts from 0 to 8. "At least" and "at most" include the boundary; "more than" and "fewer than" do not. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

Write the phrase as an inequality"at least 3" is $X \ge 3$ , "fewer than 3" is $X < 3$
Decide whether the boundary is included"at least", "at most", "no more than" and "inclusive" include it; "more than" and "fewer than" do not
For a count, turn a strict inequality into one with an equals sign $X < 3$ becomes $X \le 2$ , and $X > 4$ becomes $X \ge 5$
List the included values to checkthe slip "fewer than 3 is 3 or fewer" shows up at once

Part 5 · Key ideas

Key ideas

At least $k$ : $X \ge k$ . At most $k$ and no more than $k$ : $X \le k$ . More than $k$ : $X > k$ . Fewer than $k$ : $X < k$ . No fewer than $k$ : $X \ge k$ .
For counts, $X < k$ is $X \le k - 1$ and $X > k$ is $X \ge k + 1$ : "fewer than 3" is 0, 1 or 2.
"At least one" is every count except 0.
For continuous measurements, whether the exact boundary is included makes almost no difference.

Part 6 · Misconception

A common mistake

The wrong idea: "Fewer than 3" means $X \le 3$ .

What actually happens: "Fewer than 3" leaves out 3: it is $X < 3$ , which for counts is $X \le 2$ , the values 0, 1 and 2.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. A count X can be 0, 1, 2, and so on. Which inequality says "X is at least 3"?

$X > 3$
$X \ge 3$
$X \le 3$
$X < 3$

Show the answer

"At least 3" is 3 or more: $X \ge 3$ .

$X > 3$ : That is "more than 3", which leaves out 3.
Correct: $X \ge 3$ : Right: "at least 3" means 3 or more, so 3 itself is included.
$X \le 3$ : That is "at most 3", 3 or fewer.
$X < 3$ : That is "fewer than 3".

2. For a count X, which is the same as "fewer than 3"?

$X \le 3$
$X \ge 3$
$X < 2$
$X \le 2$

Show the answer

"Fewer than 3" means $X < 3$ . For counts, the largest such value is 2, so it is $X \le 2$ . Reading it as $X \le 3$ is a classic slip.

$X \le 3$ : That includes 3. "Fewer than 3" stops before 3.
$X \ge 3$ : That is "at least 3", the opposite group of counts.
$X < 2$ : That leaves out 2, which is fewer than 3.
Correct: $X \le 2$ : Right: fewer than 3 is $X < 3$ , and for whole-number counts that is 0, 1 or 2, so $X \le 2$ .

3. A count X can be 0 to 6. Which values are included in "at most 2"? Select all that apply.

Show the answer

"At most 2" means 2 or fewer: 0, 1 and 2.

Correct: 0: Right: at most 2 includes every count up to and including 2.
Correct: 1: Right: 1 is fewer than 2.
Correct: 2: Right: "at most" includes the boundary.
3: 3 is more than 2.
4: 4 is more than 2.
6: 6 is more than 2.

4. Out of 10 free throws, a player makes "more than 4". For the count X of free throws made, which inequality fits?

$X \ge 5$
$X \ge 4$
$X \le 4$
$X > 5$

Show the answer

"More than 4" is $X > 4$ , which for counts is $X \ge 5$ .

Correct: $X \ge 5$ : Right: more than 4 is $X > 4$ , which for whole numbers is 5 or more.
$X \ge 4$ : That includes 4, but "more than 4" does not.
$X \le 4$ : That is "at most 4".
$X > 5$ : That leaves out 5, which is more than 4.

5. A count X can be 0, 1, 2, ... , 8. Which counts make up "at least one"?

0 and 1
1 alone
1 through 8
2 through 8

Show the answer

"At least one" is $X \ge 1$ : every value except 0. It is often easiest to find by starting with all the values and removing 0.

0 and 1: Those are "at most one".
1 alone: That is "exactly one".
Correct: 1 through 8: Right: "at least one" is every count except 0.
2 through 8: That is "more than one", which leaves out 1.

6. Ten students read 0, 1, 1, 2, 3, 3, 4, 5, 6 and 9 books. How many students read at least 5 books?

Show the answer

"At least 5" includes 5: the students with 5, 6 and 9 books, so 3 students.

2: That counts only 6 and 9, leaving out the student who read exactly 5.
Correct: 3: Right: the counts 5, 6 and 9 are 5 or more.
4: That also counts the student who read 4, which is fewer than 5.
5: 5 is the boundary value, not the number of students.

Part 8 · Summary

Summary

Translate words into inequalities before computing: at least is

\ge

, at most and no more than are

\le

, more than is

>

, fewer than is

<

. For counts, change strict inequalities to include the next whole number (

X < 3

X \le 2

), and list the included values to check.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

Next: topic 2.4

Question set for this topic