Solving modulus inequalities
- A modulus inequality is really a statement about distance.
- says a is less than b from 0. So a stays inside the band .
- says the opposite: a is more than b from 0, on one side or the other.
is one band ; is two tails or . The way the sign points tells you band or tails before you work anything out.
Two pictures: one band, or two tails
Read the sign and the picture follows:
- : one band (solve it as a sandwich).
- : two separate tails (joined by “or”).
≤ : solve the sandwich in one line
Worked example
Solve
It is a band, so put the bracket between and , then work all three parts at once:
- Write the sandwich: .
- Add 5 to all three parts: .
- Halve all three parts: .
Answer
Two moduli (or a tricky one): square, then read a quadratic
- When both sides are moduli, do the same as §1.1: square to remove them, then solve the quadratic inequality you get by sketching its parabola.
- This is what the harder exam questions test.
Worked example
Solve
- Square both sides (): .
- Expand: .
- Collect to one side and factorise: .
- This upward parabola is outside its roots and .
Answer
Why squaring beats four separate cases
You could split into cases. But a modulus on both sides gives four sign combinations and lots of region checks. Squaring avoids all of that. Both sides are zero or positive, so and mean the same thing, and no fake roots appear. You get one quadratic and one parabola sketch instead of four cases.
Common mistake
Never write a case as one chain . That is impossible. always gives two answers joined by or. And if you ever divide an inequality by a negative number, flip the sign.
Now you try
Solve . (9709-style)
Your turn— tap to reveal the worked answer (9709-style)
Square both sides: , so . Outside the roots and :
Answer