Inequalities involving absolute value: |2x - 7| - 1 > 0

I need help figuring out why the answer to this question is what it is...

[2x-7]-1>0 with [ ] being absolute value

I see that 2x-7>1 so 2x >8 so x>4

but... the answer given also includes that x<3 ... How does this work?

Re: Inequalities involving absolute value

hmwin said:

I need help figuring out why the answer to this question is what it is...

[2x-7]-1>0 with [ ] being absolute value

I see that 2x-7>1 so 2x >8 so x>4

but... the answer given also includes that x<3 ... How does this work?

Click to expand...

|2x - 7| = (2x-7) and -(2x-7) ------ By definition

So

(2x-7)- 1> 0 ..............and ............... -(2x-7) - 1 > 0

2x > 8 .....................and .................... -2x > -6

x > 4 ......................and ..................... x < 3 (watch that inequality flipped - because division by negative number)

Re: Inequalities involving absolute value

Subhotosh Khan said:

|2x - 7| = (2x-7) and -(2x-7) ------ By definition

So

(2x-7)- 1> 0 ..............and ............... -(2x-7) - 1 > 0

2x > 8 .....................and .................... -2x > -6

x > 4 ......................and ..................... x < 3 (watch that inequality flipped - because division by negative number)

Click to expand...

Be careful there, Subhotosh.|x|=x if x>= 0, or |x|=-x if x<0. The imperative conjunction being "or", rather than "and". Obviously x cannot both be greater than 4 and less than 3, as your solution suggests.

Re: Inequalities involving absolute value

My solution suggests infinite solution - x is any number greater than 4 or any number less than 3 (No number between 3 and 4 - both inclusive - is in the solution set)

But your point well taken - I always get confused with the statements "and" /"or" - I also think (now) it should be "or".