Question

Сумма корней уравнения |x+2|-|x-3|+|x-1|=4 равна

Answer 1

Пожалуйста. .................

Answer 2

To find the sum of the roots of the equation |x + 2| - |x - 3| + |x - 1| = 4, we'll need to solve the equation and then find the sum of the solutions.

Let's consider different cases based on the signs of the expressions inside the absolute value:

Case 1: (x + 2) is positive, (x - 3) is positive, and (x - 1) is positive. In this case, the equation becomes: (x + 2) - (x - 3) + (x - 1) = 4

Solving this, we get: x + 2 - x + 3 + x - 1 = 4 3x + 4 = 4 3x = 0 x = 0

Case 2: (x + 2) is negative, (x - 3) is negative, and (x - 1) is positive. In this case, the equation becomes: -(x + 2) + (x - 3) + (x - 1) = 4

Solving this, we get: -x - 2 + x - 3 + x - 1 = 4 -x - 6 = 4 -x = 10 x = -10

Case 3: (x + 2) is negative, (x - 3) is positive, and (x - 1) is positive. In this case, the equation becomes: -(x + 2) - (x - 3) + (x - 1) = 4

Solving this, we get: -x - 2 - x + 3 + x - 1 = 4 -x = 4 x = -4

Case 4: (x + 2) is positive, (x - 3) is negative, and (x - 1) is positive. In this case, the equation becomes: (x + 2) + (x - 3) + (x - 1) = 4

Solving this, we get: x + 2 + x - 3 + x - 1 = 4 3x - 2 = 4 3x = 6 x = 2

Case 5: (x + 2) is positive, (x - 3) is positive, and (x - 1) is negative. In this case, the equation becomes: (x + 2) - (x - 3) - (x - 1) = 4

Solving this, we get: x + 2 - x + 3 - x + 1 = 4 2 = 4

Case 6: (x + 2) is negative, (x - 3) is negative, and (x - 1) is negative. In this case, the equation becomes: -(x + 2) + (x - 3) - (x - 1) = 4

Solving this, we get: -x - 2 + x - 3 - x + 1 = 4 -x - 4 = 4 -x = 8 x = -8

None of the solutions from Case 5 and Case 6 satisfy the equation.

So, the valid solutions are x = 0, x = -10, and x = -4. The sum of these roots is:

0 + (-10) + (-4) = -14

Therefore, the sum of the roots of the equation is -14.