Chapter 9 · Practice set 1Textbook pages 170–171

Practice Set 1 – Solutions കേവലമൂല്യം

Two table questions that lead to two important facts about absolute value: one about products, one about sums. Fill each table in your notebook first, then step through the animation.

📖 What you need (page 6 of this guide)

Question 1 · Absolute value of a product

Q1. ചുവടെയുള്ള പട്ടിക പൂർത്തിയാക്കുക. Complete the table with x, y, xy, |x|, |y|, |xy|, |x||y| for (4, 3), (−4, 3), (4, −3), (−4, −3). (i) Take more numbers and extend the table. Is there a general relation between |xy| and |x||y|? (ii) Prove that |xy| = |x||y| for any two numbers x, y.

Look at the last two columns of every row. For the proof, think about the sign of xy in each of the four sign cases of x and y.

🎬 Solution animation – the table row by row
🧪 Try any numbers
🎬 Proof animation – four sign cases
Answer: |xy| = 12 = |x||y| in every row. In general |xy| = |x||y| for all numbers: in each sign case both sides are the same positive number (and both are 0 if x or y is 0). Another proof: |xy|² = (xy)² = x²y² = |x|²|y|², and both |xy| and |x||y| are not negative.
💡 Why this matters

Absolute value "passes through" multiplication: |−3 × 7| = 3 × 7. Distances scale nicely – multiplying every number by −2 doubles every distance.

Question 2 · Absolute value of a sum

Q2. Complete the table with x, y, x + y, |x|, |y|, |x| + |y|, |x + y| for (3, 5), (−3, 5), (3, −5), (−3, −5). Take more numbers. Can you say a general relation between |x| + |y| and |x + y|?

Compare the last two columns. When are they equal? When is |x + y| smaller? Look at the signs of x and y.

🎬 Solution animation – the table row by row
🧪 Moves on the number line

Walk x from 0, then y more. |x| + |y| is the total distance walked; |x + y| is how far from 0 you end up. Walking back and forth (opposite signs) wastes distance.

Answer: rows give |x| + |y| = 8, 8, 8, 8 and |x + y| = 8, 2, 2, 8. In general |x + y| ≤ |x| + |y|: they are equal when x and y have the same sign (or one of them is 0), and |x + y| is smaller when they have opposite signs.
💡 Why this matters

This is the "triangle inequality" of the number line: the direct distance is never more than the total of two trips. It reappears in higher mathematics everywhere.

Check yourself

Next: Distances between numbers ▶