Question

|x-2|<=|x+4| ajutați-mă va rog

Answer 1

|x-2| ≤ | x+4|

|x-2| - |x+4| ≤ 0

x-2=0 =>  x=2

x+4=0 => x= -4

x     | -oo           -4                  2                   +oo

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x-2  | - - - - - - - - - - - - - - - - - 0 + + + + + + + + + +

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x-4  | - - - - - - - - 0 + + + + + + + + + + + + + + + + +

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I. x∈(-oo; -4)

|x-2| - |x+4| ≤ 0

⇔ -(x-2) + (x+4) ≤ 0

⇔ -x+2+x+4 ≤ 0

⇔ 6 ≤ 0, nu avem solutii in acest interval

II: x∈[-4; 2]

|x-2| - |x+4| ≤ 0

⇔ -(x-2) - (x+4) ≤ 0

⇔ -x+2-x-4 ≤ 0

⇔ -2x -2 ≤0

⇔ -2 ≤ 2x | :2

⇔ -1 ≤ x

cum x∈[-4; 2] ⇒ solutia in acest interval este [-1; 2]

III: x∈(2; +oo)

|x-2| - |x+4| ≤ 0

⇔ (x-2)-(x+4) ≤ 0

⇔ x-2-x-4 ≤ 0

⇔ -6 ≤ 0, deci in intervalul (2; +oo) exista solutii in intervalul (2; +oo)

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x∈[-1; 2] ∪ (2; +oo)

⇒ x∈ [-1; +oo)