|x| < 5 means the distance is under 5, so x lies between −5 and 5. That is an "and".
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Greater than
|x| > 5 means the distance exceeds 5, so x is below −5 or above 5. That is an "or".
A way to remember
Less than traps the values inside; greater than splits them to the outside.
A different centre
|x − 4| < 2 measures distance from 4, not zero, so the answer is centred on 4.
Less than gives an "and"
|x − 3| < 5 says the distance from 3 is under 5, so x is within 5 either way: −2 < x < 8. Less-than absolute value inequalities always give a single bounded interval.
Greater than gives an "or"
|x − 3| > 5 says the distance exceeds 5, so x is far out on either side: x < −2 or x > 8. Greater-than inequalities give two separate rays, never one interval.
A way to remember which is which
Less-than traps the values between two bounds; greater-than throws them apart. Thinking in terms of distance from a point makes the two cases follow from meaning rather than from a memorised rule.
The impossible and the trivial cases
|x| < −2 has no solution, since distance cannot be negative. And |x| > −2 is true for every x. Checking the sign of the right-hand side before splitting handles both cases instantly.
Step 2: Try It Yourself
Tap and try it out.
3
Step 3: Watch an Example
One step at a time.
Watch Priya Handle Both Kinds
Priya solves |x| < 4 and then |x| > 4.
- Step 1
For |x| < 4, the distance is under 4, so −4 < x < 4.
Step 4: Your Turn
Practice makes it stick.
Trapped
Problem 1 of 2
|x| < 6. What is the largest whole number solution?
Split
Problem 2 of 2
|x| > 6. What is the smallest whole number above the gap?
Inside or Outside
1 of 8
|x| < 3. Largest whole number solution?
2 of 8
|x| < 3. Smallest whole number solution?
3 of 8
|x| > 2. Smallest whole number above the gap?
4 of 8
|x − 4| < 2. What is the centre of the solution?
5 of 8
|x − 4| < 2. Largest whole number solution?
6 of 8
Match each inequality to the shape of its solution.
Tap a card on the left to start.
7 of 8
|x| > 0. Which single number fails? Give it.
8 of 8
|x| < −2. How many solutions?
Step 5: Quick Check
Show what you know.
Question 1 of 1
|x| < 8. Largest whole number solution?
What You Learned
- Less than traps the solutions into one interval around the centre.
- Greater than splits them into two intervals with a gap.
- An absolute value below a negative has no solutions at all.