Question

|2x -8| < 2

a. 3 b. -5 c. x>5 or x<3
d. x>-3 or x<-5

Answer 1

• |2x-8|<2

So

Case-1

• 2x-8<2
• 2x<2+8
• 2x<10
• x<5

Case-2

• 2x-8>-2
• 2x>6
• x>3

So solution is

• 3<x<5

Answer 2

`3 < x < 5`

Explanation:

Given inequality:

`|2x - 8| < 2`

`\textsf{Apply absolute rule}: \quad \textsf{If }|u| < a\:\textsf{ when } \: a > 0,\: \textsf{then }-a < u < a`

`\implies u=2x-8\: \textsf{ and }\:a=2`

`\implies -2 < 2x-8 < 2`

Therefore:

`\implies -2 < 2x-8`

`\implies 6 < 2x`

`\implies 3 < x`

And:

`\implies 2x-8 < 2`

`\implies 2x < 10`

`\implies x < 5`

Merge the overlapping intervals:
`3 < x < 5`

Proof

Graph:
`y=|2x-8|`

Add a line at
`y=2`

The interval that satisfies
`|2x - 8| < 2` is the area under the points of intersection (shaded area on the attached graph).

The values of x when
`y=|2x-8|` is less than 2 is more than x = 3 and less than x = 5.