Given:
|x + a| + |x| > 3, x ϵ R.
|x + a| = -(x + a) or (x + a)
|x| = -x or x
When |x + a| = -(x + a) and |x| = -x
Then,
|x + a| + |x| > 3 → -(x + a) + (-x) > 3
-x -a – x > 3
-2x – a > 3
Adding a on both the sides in above equation
-2x -a + a> 3 + a
-2x > 3 + a
Dividing both the sides by 2 in above equation
Multiplying both the sides by -1 in the above equation
Now when, |x + a| = -(x + a) and |x| = x
Then,
|x + a| + |x| > 3 → -(x + a) + x > 3
-x -a + x > 3
– a > 3
In this case no solution for x.
Now when, |x + a| = (x + a) and |x| = -x
Then,
|x + a| + |x| > 3 → (x + a) + (-x) > 3
x + a - x > 3
a > 3
In this case no solution for x.
Now when,
|x + a| = (x + a) and |x| = x
Then,
|x + a| + |x| > 3 → (x + a) + (x) > 3
x + a + x > 3
2x + a > 3
Subtracting a from both the sides in above equation
2x + a – a > 3 – a
2x > 3 – a
Dividing both the sides by 2 in above equation