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Q.

If |2x−3|<|x+5|, then x belongs to :

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a

(-3,5)

b

(5.9)

c

(\frac{- 2}{3}, 8)

d

(- 8, \frac{2}{3})

answer is C.

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Detailed Solution

Given an inequation above, which is given by |2x−3|<|x+5|,
We have to find the interval of xx, from the above inequation.
This can be in 2 ways, either solving by removing the modulus of the both the expressions in the above inequation.
Another way to do this that by squaring on both sides of the inequation, as given below:
⇒|2x−3|<|x+5|
Squaring the inequation on both sides as given below:
⇒

{| 2 x - 3 |}^{2} < {| x + 5 |}^{2}

⇒

4 x^{2} + 9 - 2 (2 x) (3) < x^{2} + 25 + 2 (5) (x)

Simplifying the expressions in the above equations, as given below:
⇒

4 x^{2} + 9 - 12 x < x^{2} + 25 + 10 x

⇒

3 x^{2} - 22 x - 16 < 0

Hence by rearranging the terms we obtained a quadratic expression, solving the quadratic expression as given below:
⇒

x < \frac{- (- 22) \pm \sqrt{{(22)}^{2} - 4 (3) (- 16)}}{2 (3)}

⇒

x < \frac{22 \pm \sqrt{676}}{6}

On further simplification, as given below:
⇒

x < \frac{22 \pm 26}{6}

⇒

x < \frac{22 + 26}{6}

;

x < \frac{22 - 26}{6}

⇒

x < 8

;

x < \frac{- 2}{3}

Hence the roots can be written as given below:
⇒(

x - 8) (x + \frac{2}{3}) < 0

∴x∈(

\frac{- 2}{3}

,8)
The interval of x belongs to (

\frac{- 2}{3}

,8)

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