Question

|x-1| + |1-2x|=2|x|. ​

Answer 1

Объяснение:

$|x-1|+|1-2x|=2*|x|.$

Найдём нули функции:

x-1=0      x=0

1-2x=0    2x=1 |÷2     x=0,5

x=0          ⇒

-∞____0____0,5____1____+∞

x∈(-∞;0].

-(x-1)+(1-2x)=2*(-x)

-x+1+1-2x=-2x

x=2 ∉.

x∈(0;0,5)

-(x-1)+(1-2x)=2x

-x+1+1-2x=2x

5x=2 |÷5

x=0,4. ∈

x∈[0,5;1) .

-(x-1)-(1-2x)=2x

-x+1-1+2x=2x

x=0. ∉

x∈[1;+∞)

x-1-(1-2x)=2*x

x-1-1+2x=2x

x=2. ∈

Ответ: x₁=0,4,  x₂=2.

Answer 2

To solve the equation |x-1| + |1-2x| = 2|x|, you can consider different cases depending on the sign of x.

Case 1: x ≥ 0 In this case, we have:

|x-1| + |1-2x| = 2x

Now, let's break this down into smaller cases:

1. If x ≥ 1: In this case, both |x-1| and |1-2x| are positive, so the equation becomes:

   (x - 1) + (2x - 1) = 2x

   Simplify it:

   3x - 2 = 2x

   Now, isolate x:

   3x - 2 - 2x = 0 x - 2 = 0 x = 2

   So, if x ≥ 1, the solution is x = 2.

2. If 0 ≤ x < 1: In this case, |x-1| is positive, and |1-2x| is negative. So, the equation becomes:

   (x - 1) - (2x - 1) = 2x

   Simplify it:

   x - 1 - 2x + 1 = 2x

   Combine like terms:

   -x = 2x

   Now, isolate x:

   -x - 2x = 0 -3x = 0 x = 0

   So, if 0 ≤ x < 1, the solution is x = 0.

Case 2: x < 0 In this case, both |x-1| and |1-2x| are negative, so we need to negate them:

-(x - 1) - (1 - 2x) = 2(-x)

Now, simplify:

-x + 1 - 1 + 2x = -2x

Combine like terms:

x - 2 = -2x

Isolate x:

x + 2x = 2 3x = 2 x = 2/3

So, if x < 0, the solution is x = 2/3.

In summary, the equation |x-1| + |1-2x| = 2|x| has solutions: x = 0, x = 2, and x = 2/3.