Write the following expression in appropriate intervals so that they are bereft of modulus sign (i) |x2 – 7x + 10| (ii) |x3 – x| (iii) |2x – 2| (iv) |x2 – 6x + 10| (v) |x – 1| + |x2 – 3x + 2| (vi) 2 x 6x 9 (vii) 2 (x–1) + |x + 2| – 3 |x+1|
Answers
Given : (i) |x² – 7x + 10|
(ii) |x³ – x|
(iii) |2x – 2|
To Find : Write the expression in appropriate intervals without modulus sign
Solution:
| x | = x if x ≥ 0
= - x if x < 0
(i) |x² – 7x + 10|
= | (x - 2)(x - 5) |
(x - 2)(x - 5) ≥ 0 if x ≥ 5 or x ≤ 2
(x - 2)(x - 5) < 0 if 2 < x < 5
Hence
x² – 7x + 10 x ≤ 2
- ( x² – 7x + 10 ) 2 < x < 5
x² – 7x + 10 x ≥ 5
|x³ – x|
= | x ( x² - 1) |
x ≥ 1 => x ( x² - 1) ≥ 0
x < - 1 => x ( x² - 1) < 0
-1 ≤ x ≤ 0 => x ( x² - 1) ≥ 0
0 < x < 1 => x ( x² - 1) < 0
Hence
x - x³ if x < - 1
x³ – x if -1 ≤ x ≤ 0
x - x³ if 0 < x < 1
x³ – x if x ≥ 1
|2x – 2|
2x - 2 if x ≥ 1
2 - 2x if x < 1
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