An "and" inequality needs both parts true at once, so the answer is where they overlap.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Or
An "or" inequality needs only one part true, so the answer is everything either one covers.
The between form
3 < x < 8 is an "and" written compactly. It means x beats 3 and is beaten by 8.
And can be empty
x > 5 and x < 2 overlap nowhere, so nothing satisfies it.
And means both, or means either
A compound inequality joins two conditions. "And" requires both to hold, giving the overlap of two ranges. "Or" requires at least one, giving the union. Which word appears changes the answer entirely.
The compact form
−2 < x < 5 is shorthand for x > −2 and x < 5. This notation is only used for "and" statements; an "or" solution cannot be written this way and must be given as two separate pieces.
Operate on all three parts
To solve −2 < 2x + 4 < 10, subtract 4 from all three parts, then divide all three by 2. Whatever you do must be done everywhere, and if you divide by a negative both symbols flip.
An "and" can be empty
x > 5 and x < 2 has no solution, since no number is both. An "or" of the same two conditions covers almost everything. Sketching both ranges on one line makes the outcome obvious before you write anything.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Priya Solve a Between
Priya solves 1 < x + 2 < 7.
- Step 1
She subtracts 2 from all three parts at once.
Step 4: Your Turn
Practice makes it stick.
The Overlap
Problem 1 of 2
x > 3 and x < 9. What is the smallest whole number that works?
The Empty Case
Problem 2 of 2
x > 8 and x < 3. How many numbers satisfy it?
And or Or
1 of 8
2 < x < 6. Smallest whole number?
2 of 8
2 < x < 6. Largest whole number?
3 of 8
0 < x + 1 < 5. Largest whole number for x?
4 of 8
x < 1 or x > 9. Does x = 5 work? 1 for yes, 0 for no.
5 of 8
x < 1 or x > 9. Does x = 10 work? 1 for yes, 0 for no.
6 of 8
Sort each compound inequality by whether anything satisfies it.
Tap something to move it.
- Empty
- Empty
7 of 8
−3 < x < 3. How many whole numbers satisfy it?
8 of 8
Can an "or" inequality ever be empty? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
4 < x < 10. Smallest whole number?
What You Learned
- An "and" inequality is satisfied only where both parts overlap.
- An "or" inequality is satisfied by anything either part covers.
- An "and" can be empty; an "or" never is.