Question

Помогите решить уравнения 1) |x|-х=2 2) (х^2)-5|x|+6=0 3) |3x+2|=11-х 4) |(x^2)-9|=9-(x^2)

Answer 1

1) lxl=x+2,  x+2>=0,  x>=-2,   x=x+2  или  x=-x-2,    0 не=2,  2x=-2,  x=-1

2)x>=0, то  x^2-5x+6=0, корни  2 и 3,  если х<0, то x^2+5x+6=0,   корни -2 и -3

3)  11-x>=0,  x<=11,   3x+2=11-x  или  3x+2=x-11,    4x=9  или  2x=-13,   x=9/4  или  x=-13/2

4) можно подбором  х=3  или х=-3

Answer 2

1. |x|-x=2

We can solve this equation by considering two cases, one for x >= 0 and one for x < 0.

Case 1: x >= 0 In this case, |x| = x, so the equation becomes: x - x = 2 which simplifies to 0 = 2, which is not true. Therefore, there are no solutions in this case.

Case 2: x < 0 In this case, |x| = -x, so the equation becomes: -x - x = 2 which simplifies to -2x = 2, or x = -1. This is a valid solution since |-1| - (-1) = 1 + 1 = 2.

Therefore, the only solution to the equation |x|-x=2 is x = -1.

2. x^2 - 5|x| + 6 = 0

We can solve this equation by considering two cases, one for x >= 0 and one for x < 0.

Case 1: x >= 0 In this case, |x| = x, so the equation becomes: x^2 - 5x + 6 = 0 which factors as (x-3)(x-2) = 0. Therefore, the solutions in this case are x = 2 and x = 3.

Case 2: x < 0 In this case, |x| = -x, so the equation becomes: x^2 + 5x + 6 = 0 which factors as (x+3)(x+2) = 0. Therefore, the solutions in this case are x = -2 and x = -3.

Therefore, the solutions to the equation x^2 - 5|x| + 6 = 0 are x = -3, -2, 2, and 3.

3. |3x+2| = 11-x

We can solve this equation by considering two cases, one for 3x+2 >= 0 and one for 3x+2 < 0.

Case 1: 3x+2 >= 0 In this case, |3x+2| = 3x+2, so the equation becomes: 3x+2 = 11-x which simplifies to 4x = 9, or x = 9/4. This is a valid solution since |3(9/4)+2| = 11-(9/4).

Case 2: 3x+2 < 0 In this case, |3x+2| = -(3x+2), so the equation becomes: -(3x+2) = 11-x which simplifies to -3x-13 = 0, or x = -13/3. This is a valid solution since |-3(-13/3)+2| = 11-(-13/3).

Therefore, the solutions to the equation |3x+2| = 11-x are x = -13/3 and x = 9/4.

4. |x^2-9| = 9-(x^2)

We can solve this equation by considering two cases, one for x^2-9 >= 0 and one for x^2-9 < 0.

Case 1: x^2-9 >= 0 In this case, |x^2-9| = x^2-9, so the equation becomes: x^2-9 = 9-x^2 which simplifies to 2x^2 = 18