Question

|2x-3|-|4-x|=x-1 помогите |- это модуль

Answer 1

To solve the equation |2x-3|-|4-x|=x-1, we need to consider the different cases based on the values inside the absolute value signs. Let's break it down step by step:

Case 1: (2x-3) and (4-x) are both positive

In this case, we can remove the absolute value signs without changing the equation. The equation becomes: (2x-3) - (4-x) = x-1

Simplifying the equation: 2x - 3 - 4 + x = x - 1 3x - 7 = x - 1

Now, let's solve for x: 3x - x = 1 - 7 2x = -6 x = -3

Case 2: (2x-3) is positive and (4-x) is negative

In this case, we need to change the sign of (4-x) when removing the absolute value sign. The equation becomes: (2x-3) + (x-4) = x-1

Simplifying the equation: 2x - 3 + x - 4 = x - 1 3x - 7 = x - 1

Now, let's solve for x: 3x - x = 1 - 7 2x = -6 x = -3

Case 3: (2x-3) is negative and (4-x) is positive

Similar to Case 2, we need to change the sign of (2x-3) when removing the absolute value sign. The equation becomes: -(2x-3) - (4-x) = x-1

Simplifying the equation: -2x + 3 - 4 + x = x - 1 -x - 1 = x - 1

In this case, the equation simplifies to -x - 1 = x - 1. This means that x can be any real number since the equation is always true.

Case 4: (2x-3) and (4-x) are both negative

In this case, we need to change the signs of both (2x-3) and (4-x) when removing the absolute value signs. The equation becomes: -(2x-3) + (4-x) = x-1

Simplifying the equation: -2x + 3 + 4 - x = x - 1 -3x + 7 = x - 1

Now, let's solve for x: -3x - x = 1 - 7 -4x = -6 x = 3/2

Summary of Solutions:

From the different cases, we have found two possible solutions for the equation |2x-3|-|4-x|=x-1: - x = -3 - x = 3/2

Please note that these solutions should be verified by substituting them back into the original equation to ensure they satisfy the equation.

I hope this helps! Let me know if you have any further questions.