Question

Rozwiąż nierówność: |x+3|-|x-1|>1

Answer 1

|x+3|-|x-1|>1

 

  I         II        III

------o-------o--------

       -3        1

 

I x ∈ (-oo, - 3) ⇒ -(x + 3) - [ -(x - 1) ] > 1

                           -x - 3 + x - 1 > 1

                              0 - 4 > 1

                              0 > 5  (fałsz)

                              x ∈ {Ф}

          x ∈ {Ф}

II x ∈ <- 3, 1) ⇒ (x + 3) - [ -(x - 1) ] > 1

                           x + 3 + x - 1 > 1

                              2x + 2 > 1

                              2x >  -1

                              x > -1/2

         x ∈ (-1/2, 1)

 

III   x ∈ <1, +oo) ⇒ (x + 3) - (x - 1) ] > 1

                              x + 3 - x + 1 > 1

                                     4 > 1  (prawda)

                                      x ∈ R

         x ∈ <1, +oo)

 

odp. x ∈ (-1/2, +oo)