|2 - x| =

{x - 2 , x ≥ 2
{ 2 - x , x < 2



|2x + 1| =

{ 2x + 1, x ≥ -½
{ -2x - 1 , x < -½


The inequality is given based in convention. It will vary depending on the convention your lecturer sets

(i.e which piece of the function to be excluded for x = 2 and x = -½)

(i) -½
(ii) 2
(iii) 2 - x
(iv) x - 2
(v) -2x - 1
(vi) 2x + 1

So,


① For x < -½ ,

|2 - x| - |2x + 1|

= 2 - x - (-2x - 1)

= 2 - x + 2x + 1

= x + 3


When |2 - x| - |2x + 1| > 0,

x + 3 > 0
x > -3

But we have to consider the domain so -3 < x < -½


② For -½ ≤ x < 2,

|2 - x| - |2x + 1|

= 2 - x - (2x + 1)

= 2 - x - 2x - 1

= 1 - 3x


When |2 - x| - |2x + 1| > 0,

1 - 3x > 0
1 > 3x
x < ⅓

But we have to consider the domain so -½ ≤ x < ⅓


③ For x ≥ 2,

|2 - x| - |2x + 1|

= x - 2 - (2x + 1)

= x - 2 - 2x - 1

= -3 - x


When |2 - x| - |2x + 1| > 0,

-3 - x > 0

-3 > x

x < -3
(not applicable as x ≥ 2 for this portion of the function. It cannot be extrapolated backwards.)



So -3 < x < -½ and -½ ≤ x < ⅓

Combining the inequalities,

-3 < x < ⅓