Question

|x^2-4x+3|+2<2|x-1|+|x+3|​

Answer 1

Ответ:

Объяснение:

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Answer 2

To solve the inequality |x^2-4x+3| + 2 < 2|x-1| + |x+3|, we need to consider different cases. Let's analyze them one by one.

First, consider x ≤ 1. In this case, the absolute values in the inequality change as follows:

• |x^2-4x+3| becomes |x^2-4x+3| = x^2 - 4x + 3 (since x^2-4x+3 is already non-negative for x ≤ 1).
• |x-1| becomes |x-1| = 1 - x.
• |x+3| becomes |x+3| = -(x + 3) (since x + 3 is non-positive for x ≤ 1).

So the inequality becomes:

x^2 - 4x + 3 + 2 < 2(1 - x) - (x + 3)

Simplify this to get:

x^2 - 4x + 5 < 2 - 2x - x - 3

Simplify further:

x^2 - 4x + 5 < -3x - 1

Rearrange to get a quadratic inequality:

x^2 + x - 4x + 5 + 3x + 1 < 0

x^2 - 2x + 6 < 0

The discriminant of this quadratic is (-2)^2 - 416 = 4 - 24 = -20, which is negative, indicating that the quadratic is always positive. Therefore, this case does not provide any solution.

Next, consider 1 < x < 3. In this case, the absolute values change as follows:

• |x^2-4x+3| becomes |x^2-4x+3| = x^2 - 4x + 3 (since x^2-4x+3 is already non-negative for 1 < x < 3).
• |x-1| becomes |x-1| = x - 1.
• |x+3| becomes |x+3| = x + 3.

So the inequality becomes:

x^2 - 4x + 3 + 2 < 2(x - 1) + (x + 3)

Simplify this to get:

x^2 - 4x + 5 < 2x - 2 + x + 3

Simplify further:

x^2 - 4x + 5 < 3x + 1

Rearrange to get a quadratic inequality:

x^2 - 7x + 4 < 0

Factor this to get:

(x - 4)(x - 1) < 0

This inequality is satisfied when 1 < x < 4.

Lastly, consider x ≥ 3. In this case, the absolute values change as follows:

• |x^2-4x+3| becomes |x^2-4x+3| = x^2 - 4x + 3 (since x^2-4x+3 is already non-negative for x ≥ 3).
• |x-1| becomes |x-1| = x - 1.
• |x+3| becomes |x+3| = x + 3.

So the inequality becomes:

x^2 - 4x + 3 + 2 < 2(x - 1) + (x + 3)

Simplify this to get:

x^2 - 4x + 5 < 2x - 2 + x + 3

Simplify further:

x^2 - 4x + 5 < 3x + 1

Rearrange to get a quadratic inequality:

x^2 - 7x + 4 < 0

Factor this to get:

(x - 4)(x - 1) < 0

This inequality is satisfied when 1 < x < 4.

Therefore, the solution to the inequality |x^2-4x+3| + 2 < 2|x-1| + |x+3| is 1 < x < 4.