Chapter 9 · Section 4Textbook pages 171–174

Distances അകലങ്ങൾ

The distance from 0 to x is |x|. What about the distance between any two numbers? Recall the change of position from chapter 7: the distance is just that change with its sign thrown away. That gives a short formula, |x − y|, and a neat way to solve equations like |x − 1| = 3.

📖 What you already know

The change of position from x to y is y − x: positive for a move right, negative for a move left (chapter 7, സ്ഥാനമാറ്റം).
|x| is the distance of x from 0, and |−x| = |x| (page 6).

1. Change of position and distance

🎬 Animation: the six moves of page 171

A move to the right: the change of position is the distance. A move to the left: the change is the distance made negative. Either way, the distance is the absolute value of the change of position.

⭐ Key box (textbook page 172)

The change from x to y is y − x, so the distance between x and y is |y − x|. Since −(y − x) = x − y and a number and its negative have the same absolute value, |y − x| = |x − y|.

സംഖ്യാരേഖയിൽ x, y എന്നീ സംഖ്യകൾ തമ്മിലുള്ള അകലം |x − y|
The distance between the numbers x and y on the number line is |x − y|.

2. Bigger minus smaller

If x is bigger, x − y is positive and |x − y| = x − y. If y is bigger, x − y is negative and |x − y| = −(x − y) = y − x.

⭐ Key box (textbook page 172)

സംഖ്യാരേഖയിൽ രണ്ടു സംഖ്യകൾ തമ്മിലുള്ള അകലം, വലിയ സംഖ്യയിൽനിന്നു ചെറിയ സംഖ്യ കുറച്ചു കിട്ടുന്ന സംഖ്യയാണ്.
The distance between two numbers is the bigger minus the smaller. So |x − y| is the algebraic form of "subtract the smaller from the bigger".

🎬 Animation: 4½ to 6¼, and 4½ to −6¼

🎚️ Distance between any two numbers

x y

3. Solving |x − 1| = 3

🎬 Animation: the two numbers at distance 3 from 1

💡 The algebra way

If x > 1, |x − 1| = x − 1, and x − 1 = 3 gives x = 4. If x < 1, |x − 1| = 1 − x, and 1 − x = 3 gives x = 1 − 3 = −2. Same two answers.

4. Solving |x + 1| = 3

🎬 Animation: turn the sum into a difference

5. Inequalities: |x − 1| < 3

🎬 Animation: all the numbers closer than 3 to 1

📜 Side box: smaller and bigger ചെറുതും വലുതും (page 174)

|x − 3| < 5 means x is less than 5 from 3: −2 < x < 8. And |x − 3| > 5? x is more than 5 to the right of 3, or more than 5 to the left: x < −2 or x > 8 – two rays pointing away.

🧪 Solve |x − a| = r, < r or > r

a r type

🌍 Real life: tolerances

A machine part must be 10 mm long, allowed to be off by at most 0.2 mm. A length x is acceptable when |x − 10| ≤ 0.2, that is 9.8 ≤ x ≤ 10.2. Engineers, pharmacists and quality inspectors write limits like this every day.

6. Check yourself

🧭 Summary

Distance = |change of position|. The distance between x and y is |x − y| = |y − x| = bigger − smaller.
|x − a| = r: the two numbers a + r and a − r. |x + a| = |x − (−a)| is the distance from −a.
|x − a| < r: a − r < x < a + r. |x − a| > r: x < a − r or x > a + r.

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