Chapter 9 · Practice set 2Textbook page 174

Practice Set 2 – Solutions അകലങ്ങൾ

Three questions, each with four parts, all built on one idea: |x − a| is the distance between x and a on the number line. "= r" means the two points at distance r from a; "< r" means everything closer than r.

📖 What you need (page 8 of this guide)

The distance between x and y on the number line is |x − y|; the distance of x from 0 is |x|.
|x + 2| = |x − (−2)| is the distance between x and −2.
|x − a| < r ⇔ a − r < x < a + r (closer to a than r).

Question 1 · Solve the equations

Q1. Find x in each equation: (i) |x| = 5 (ii) |x − 3| = 2 (iii) |x − 2| = 3 (iv) |x + 2| = 3 ചുവടെപ്പറയുന്ന ഓരോ സമവാക്യവും ശരിയാകുന്ന x കണക്കാക്കുക.

Read |x − a| = r as "x is at distance r from a". There are two such points: a + r to the right and a − r to the left. For |x + 2|, write it as |x − (−2)|.

🎬 Solution animation

Answer

(i) x = 5 or −5(ii) x = 5 or 1(iii) x = 5 or −1(iv) x = 1 or −5

💡 Why two answers?

A distance does not say which side. Every equation |x − a| = r with r > 0 has exactly two solutions, a − r and a + r, one on each side of a. (If r = 0 there is only one, x = a.)

Question 2 · Between which numbers?

Q2. For each statement, find the numbers between which x must lie: (i) |x| < 3 (ii) |x − 2| < 1 (iii) |x − 1| < 2 (iv) |x + 1| < 2 ചുവടെപ്പറയുന്ന ഓരോ വാക്യവും ശരിയാകാൻ x ഏതൊക്കെ സംഖ്യകൾക്ക് ഇടയിലായിരിക്കണം എന്നു കണക്കാക്കുക.

"Distance from a less than r": mark a, go r to the left and r to the right. x lies strictly between those two points (the end points themselves are not included).

🎬 Solution animation

Answer

(i) −3 < x < 3(ii) 1 < x < 3(iii) −1 < x < 3(iv) −3 < x < 1

💡 Open circles

The ends are drawn as hollow circles because "<" excludes them: in (ii) x = 3 gives |3 − 2| = 1, which is not less than 1.

Question 3 · The integers inside

Q3. Find the integers for which each statement of Question 2 is true. രണ്ടാമത്തെ കണക്കിലെ വാക്യങ്ങൾ ഓരോന്നും ശരിയാകുന്ന പൂർണ്ണസംഖ്യകൾ ഏതൊക്കെയാണെന്ന് കണക്കാക്കുക.

Take each interval from Question 2 and list the whole numbers (positive, negative or 0) strictly inside it. Leave out the end points.

🎬 Solution animation

Answer

(i) −2, −1, 0, 1, 2(ii) 2(iii) 0, 1, 2(iv) −2, −1, 0

🌍 Why this matters

Engineers write tolerances exactly like this: a bolt of length "40 mm ± 0.5 mm" must satisfy |x − 40| ≤ 0.5. A quality checker who knows that x lies between 39.5 and 40.5 can sort bolts in seconds.

Try your own

🎚️ |x − a| = r and |x − a| < r

a r

Check yourself

🧭 Summary

|x − a| = r ⇒ x = a + r or x = a − r (two points at distance r from a).
|x − a| < r ⇒ a − r < x < a + r (an open interval around a).
For |x + b| use |x − (−b)|: the centre is −b.

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