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प्रश्न
Find all values of x that satisfies the inequality `(2x - 3)/((x - 2)(x - 4)) < 0`
उत्तर
The given inequality is f(x) = `(2x - 3)/((x - 2)(x - 4))`
= `(2(x - 3/2))/((x - 2)(x - 4)) < 0`
[The critical numbers of f(x) are those values of x for which f(x) = 0, and those values of x for which f(x) is not defined. When x = 2, f(x) = `oo` ⇒ f(x) is not defined.]
The critical numbers are x = `3/2`, x = 2 , x = 4
Divide the number into 4 intervals
`(- oo, 3/2)`, (3/2, 2), (2, 4), `(4, oo)`
(i) `(- oo, 3/2)`
When `x < 2/3` say x = 0
The factor `x - 3/2 = 0 - 3/2 < 0`
The factor x – 2 = 0 – 2 < 0
The factor x – 4 = 0 – 4 < 0
Thus `x - 3/2 < 0,x - 2< 0` and x – 4 < 0
∴ `(2(x - 3/2))/((x - 2)(x - 4)) < 0`
Thus `(2x - 3)/((x - 2)(x - 4)) <0` is true in the inteval `(- oo, 3/2)`
∴ It has solution in `(- oo, 3/2)`.
(ii) `(3/2, 2)`
When `3/2 < x< 2` say x = `(3/2 + 2)/2`
= `(3 + 4)/4`
= `7/4`
The factor `x - 3/2 = 7/4 - 3/2 = (7 - 6)/4 = 1/4 > 0`
The factor `x - 2 = 7/4 - 2 = (7 - 8)/4 = - 1/4 < 0`
The factor `x - 4 = 7/4 - 4 = (7 - 16)/4 = - 9/4 < 0`
Thus `x - 3/2 > 0, x - 2 < 0` and x – 4 < 0
∴ `(2(x - 3/2))/((x - 2)(x - 4)) > 0`
Thus `(2x - 3)/((x - 2)(x - 4)) < 0` is not true in the inteval `(3/2, 2)`
∴ It has no solution in the interal `(3/2, 2)`.
(iii) (2, 4)
When 2 < x < 4 say x = 3
The factor `x - 3/2 = 3 - 3/2 = 3/2 > 0`
The factor x – 2 = 3 – 2 = 1 > 0
The factor x – 4 = 3 – 4 = – 1 < 0
Thus `x - 3/2 > 0, x - 2 > 0` and x – 4 < 0
∴ `(2(x - 3/2))/((x - 2)(x - 4)) < 0`
Thus `(2x - 3)/((x - 2)(x - 4) < 0` is true in the interval (2, 4)
∴ It has solution in (2, 4).
(iv) `(4, oo)`
When x > 4 say x = 5
The factor `x - 3/2 = 5 - 3/2 = 7/2 > 0`
The factor x – 2 = 5 – 2 = 3 > 0
The factor x – 4 = 5 – 4 = 1 > 0
Thus `x - 3/2 > 0, x - 2 > 0` and x – 4 > 0
∴ `(2(x - 3/2))/((x - 2)(x - 4)) > 0`
Thus `(2x - 3)/((x - 2)(x - 4) < 0` is not true in the interval `(4, oo)`
∴ It has a solution in `(4, oo)`.
| Interval | Sign of `x - 3/2` |
Sign of x – 2 |
Sign of x – 4 |
Sign of `(2(x - 3/2))/((x - 2)(x - 4))` |
| `(- oo, 3/2)` | – | – | – | – |
| `(3/2, 2)` | + | – | – | + |
| (2, 4) | + | + | – | – |
| `(4, oo)` | + | + | + | + |
Hence, the inequality `(2x - 3)/((x - 2)(x - 4)) < 0` hs soluion in the interval `(- oo, 3/2)` and (2, 4).
∴ The solution set is `(- oo, 3/2)` ∪ (2, 4)
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