Inequalities · Apply · 4.3

Compound inequalities.

A compound inequality joins two conditions. “And” needs BOTH true — the overlap, like 2 < x < 7. “Or” needs EITHER true — the union. A value solves an “and” only if it satisfies both parts at once.

01 / THE IDEA

See it work.

Watch the method

Compound inequalities

“And” = overlap (both true); “or” = union (either true).

1 / 1
x>2 and x<7\text{x>2 and x<7}
1

And vs Or

“And” means both must hold (the overlap); “or” means either (the union).

x>2andx<7x>2 and x<7
— / WHY IT WORKS

The idea underneath.

Both vs either

For “and”, test a number against both inequalities; it must pass both.

Why it works

2 < x < 7 means between 2 and 7.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Which number satisfies BOTH x > 3 and x < 8?

1

Answer

6

EX 2medium

Which number satisfies BOTH x > 4 and x < 7?

1

Answer

6

EX 3hard

Which number satisfies BOTH x > 4 and x < 8?

1

Answer

7

04 / COMMON TRAPS

Where students slip.

Trap

Treating “and” like “or”.

Fix

“And” needs BOTH conditions, not just one.

Trap

Including the boundary on a strict inequality.

Fix

x > 2 excludes 2 itself.

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

Which number satisfies BOTH x > 1 and x < 6?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
C2

Which number satisfies BOTH x > 1 and x < 4?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
C3

Which number satisfies BOTH x > 3 and x < 7?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
C4

Which number satisfies BOTH x > 4 and x < 8?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
Apply

Apply (4)

A1

Which number satisfies BOTH x > 2 and x < 5?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
A2

Which number satisfies BOTH x > 4 and x < 8?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
A3

Which number satisfies BOTH x > 2 and x < 7?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
A4

Which number satisfies BOTH x > 4 and x < 8?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
Extension

Extension (4)

E1

Which number satisfies BOTH x > 3 and x < 8?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
E2

Which number satisfies BOTH x > 2 and x < 5?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
E3

Which number satisfies BOTH x > 3 and x < 8?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.
E4

Which number satisfies BOTH x > 1 and x < 6?

Need a hint?
  • “And” → the number must satisfy both.
  • Check it against each part.