Question

Find all values of x that satisfies the inequality `(2x - 3)/((x - 2)(x - 4)) < 0`

Answer 1

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)