Inequalities · Apply · 4.1

At least / at most.

Real limits — “at least”, “at most”, “no more than”, “minimum”, “maximum” — translate to ≤ and ≥. Naming the quantity x and choosing the inclusive symbol models the constraint exactly.

01 / THE IDEA

See it work.

Watch the method

At least / at most

“At most 9” → x ≤ 9; “at least 9” → x ≥ 9.

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a number is at most 9\text{a number is at most 9}
1

Translate the phrase

Pick the symbol, then write x and the number.

x≤9x ≤ 9
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Solve for x.

Type the solution, e.g. x < 4 (you can write <= for ≤).

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— / WHY IT WORKS

The idea underneath.

Model the limit

A maximum is a ≤ bound; a minimum is a ≥ bound — both include the boundary.

Why it works

“No more than 9” still allows 9.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Write this as an inequality (use x): “a number is less than 12”.

1

Translate the phrase

Pick the symbol, then write x and the number. x < 12

EX 2medium

Write this as an inequality (use x): “a number is less than 12”.

1

Translate the phrase

Pick the symbol, then write x and the number. x < 12

EX 3hard

Write this as an inequality (use x): “a number is at least 11”.

1

Translate the phrase

Pick the symbol, then write x and the number. x ≥ 11

04 / COMMON TRAPS

Where students slip.

Trap

Using < for “at most”.

Fix

“At most” includes the value → ≤.

Trap

Swapping minimum and maximum.

Fix

Minimum → ≥; maximum → ≤.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Write this as an inequality (use x): “a number is at most 8”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
C2

Write this as an inequality (use x): “a number is less than 12”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
C3

Write this as an inequality (use x): “a number is less than 4”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
C4

Write this as an inequality (use x): “a number is more than 9”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
Apply

Apply (4)

A1

Write this as an inequality (use x): “a number is at least 11”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
A2

Write this as an inequality (use x): “a number is at least 2”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
A3

Write this as an inequality (use x): “a number is less than 6”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
A4

Write this as an inequality (use x): “a number is less than 8”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
Extension

Extension (4)

E1

Write this as an inequality (use x): “a number is at least 12”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
E2

Write this as an inequality (use x): “a number is at most 3”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
E3

Write this as an inequality (use x): “a number is less than 12”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.
E4

Write this as an inequality (use x): “a number is less than 9”.

Need a hint?
  • Maximum/“at most” → ≤.
  • Minimum/“at least” → ≥.