Page 14 - mathsvol1ch1to3ans

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4. Find two irrational numbers such that their sum is a rational number. Can you find two irrational
numbers whose product is a rational number.
Solution: As discussed in the previous example, the sum of two irrational numbers can be rational
√
√
number. Example : 3 + 2, 3 − 2. The product of these two numbers is rational number. Note
that if x and y are irrational numbers than atleast one of x + y, x − y is irrational number.
1
5. Find a positive number smaller than . Justify.
2 1000
1 1 1 1
Solution: is a positive number smaller than . Justification: Let > . Multiply
2 1001 2 1000 2 1001 2 1000
both sides by 2 1001 . We have 1 > 2 which is a contradiction. Hence the result.

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Exercise - 2.7

1. Solve for x:
3 1

(i) |3 − x| < 7. (ii) |4x − 5| ≥ −2. (iii) 3 − x ≤ . (iv) |x| − 10 < −3

4 4

1 1 3

(v) x − < x − .

4 2 4
Solution:
(i) |3 − x| < 7.
−7 < (3 − x) < 7
⇒ 7 > (x − 3) > −7
⇒ 10 > x > −4
⇒ −4 < x < 10

(ii) |4x − 5| ≥ −2.
Since LHS is always positive and RHS has negative integer, the values of x is R.

3 1

(iii) 3 − x ≤ .

4 4

1 3 1
− ≤ 3 − x ≤
4 4 4

1 3 1
≥ x − 3 ≥ −
4 4 4

13 3 11
≥ x ≥
4 4 4
11 13
≤ x ≤
3 3
(iv) |x| − 10 < −3.
−7 < x < 7

1 1 3

(v) x − < x − .

4 2 4