Ayush1692 wrote:

Is |x| ≤ 4 ?

(1) |x + 2| + |x − 4| > 2|x − 1|

(2) |(x + 4)(x − 3)| > 8

I will use other visual approach

(1) |x + 2| + |x − 4| > 2|x − 1|Critical points -2, 1, 4

Representing on number line

-----Invalid----

-2--

+++----

1----

+++-------

4----Invalid--------

Now let's examine each region

x\leq{-2} :

Let x = -10..........|-8| + |-14| > 2|-11|..........22 > 22.......Invalid.............Hence NO number satisfy the region.

-2<x<1:

Let x = 0..............|2| + |-4| > 2|-1|..........6 > 4.........Hence every Number satisfies the region...........Answer Is Yes

1<x<4:

Let x = 3..............|5| + |-1| > 2|2|..........6 > 4.........Hence every Number satisfies the region............Answer Is Yes

x\geq{4}

Let x = 10..........|12| + |6| > 2|-9|..........18 > 18.......Invalid.............Hence NO number satisfy the region.

From above:

Every x in region

-2<x<4 satisfies the statement 1 and |x| also less than 4...........Answer is always yes

Sufficient

(2) |(x + 4)(x − 3)| > 8critical points: -4 & 3

------

++++------

-4--

++++-------

3----

+++--------

x\leq{-4} :

Let x = -10..........|6 * -13 | > 8....................Answer is NO

-4<x<3:

Let x = 0..........|4 * -3 | > 8........................Answer is Yes

We can stop here as two different answers but I will continue the last region for clarification

X >3

Let x = 10..........|14* 7 | > 8......................Answer is NO.

Insufficient

Answer: A