Consider the following inequalities for the sides of any triangle ABC: (1) AC − AB < BC (2) BC − AC < AB (3) AB − BC < AC. Which of the above inequalities are always true for a valid triangle?

Introduction / Context:
This question is based on the triangle inequality, a fundamental property of any triangle. It tests whether you understand not only that each side is less than the sum of the other two, but also that the difference between any two sides is less than the third side.

Given Data / Assumptions:

• We have a triangle ABC with side lengths AB, BC and AC.
• Three inequalities are given: (1) AC − AB < BC (2) BC − AC < AB (3) AB − BC < AC
• All side lengths are positive real numbers.
• We assume ABC is a valid triangle in Euclidean geometry.

Concept / Approach:
For any triangle with sides x, y and z, the triangle inequality states: x + y > z y + z > x z + x > y From these inequalities we can also derive that the absolute difference between any two sides is less than the third side: |x − y| < z and similarly for other pairs. Each of the given inequalities is essentially one form of such a difference inequality, but without absolute values they still hold because if the left side becomes negative, it is automatically less than a positive side length.

Step-by-Step Solution:
Step 1: Consider inequality (1): AC − AB < BC. Step 2: Starting from the triangle inequality AC < AB + BC, subtract AB from both sides: AC − AB < BC. So (1) is always true. Step 3: Consider inequality (2): BC − AC < AB. Step 4: From BC < AC + AB, subtract AC from both sides: BC − AC < AB. So (2) is always true. Step 5: Consider inequality (3): AB − BC < AC. Step 6: From AB < BC + AC, subtract BC from both sides: AB − BC < AC. So (3) is always true. Step 7: In cases where the difference is negative, such as AC − AB being negative, the inequality remains valid because a negative number is automatically less than any positive side length. Step 8: Therefore all three inequalities (1), (2) and (3) hold for every valid triangle ABC.

Verification / Alternative check:
You can test with a concrete example, such as a 3 4 5 triangle. Let AB = 3, BC = 4 and AC = 5. Then: AC − AB = 5 − 3 = 2 which is less than BC = 4, so (1) holds. BC − AC = 4 − 5 = −1 which is less than AB = 3, so (2) holds. AB − BC = 3 − 4 = −1 which is less than AC = 5, so (3) holds. This numerical check supports the algebraic reasoning.

Why Other Options Are Wrong:
1 and 2 only, 2 and 3 only, 1 and 3 only: Each of these options wrongly excludes one of the inequalities, even though all are valid consequences of the triangle inequality.
None of these: This would imply none of the combinations are correct, which contradicts the derivation showing that all three are true.

Common Pitfalls:
Learners sometimes believe that the triangle inequality only provides information about sums of sides, not differences. Another mistake is to worry about the sign of expressions like BC − AC and think a negative result invalidates the inequality. Remember that a negative value is always less than a positive side length, so the inequality still holds.

Final Answer:
All three inequalities are always true, so the correct choice is 1, 2 and 3.