Inside = +c or −c: |something| = c (with c ≥ 0) means the inside is +c or −c — two equations to solve.
Always check c ≥ 0 first: |…| can never equal a negative number.
IB-style question — modulus equation
Solve |2x − 1| = 5.
Step by step
- The inside is +5 or −5.
- Solve each.
Final answer
x = 3 or x = −2.
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Less than = inside a band; greater than = outside: |x| < a ⟺ −a < x < a (a band around 0).
|x| > a ⟺ x < −a or x > a (everything outside).
Replace x by the inside expression and solve.
IB-style question — 'less than'
Solve |x − 2| < 3.
Step by step
- 'Less than' → a band: −3 < inside < 3.
- Add 2 throughout.
Final answer
−1 < x < 5.
IB-style question — 'greater than'
Solve |2x + 1| ≥ 4.
Step by step
- 'Greater than' → outside: inside ≥ 4 or inside ≤ −4.
- Solve each.
Final answer
x ≥ 3/2 or x ≤ −5/2.