Question

1.||x-1|-1|=|x-2|2.|2x-2|-|x|≥x​

Answer 1

y=2|x-1| -график синего цвета

a)x-1≥0; x≥1 график  y=2(x-1)

б)x-1<0; x<1 график y=2(1-x)

y=x+|x|-график розового цвета

a)x≥0; y=x+x=2x

б)x<0; y=x-x=0

синяя линия не ниже розовой на интервале x=(-∞;0.5]

Answer 2

Let's solve these two absolute value inequalities step by step:

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

To solve this equation, we'll consider two cases based on the sign of $x-1$:

Case 1: $x-1$ is non-negative ($x \geq 1$): In this case, the equation becomes: $x-1-1=x-2$ Simplify: $x-2=x-2$ Now, we see that this equation holds true for all $x \geq 1$.

Case 2: $x-1$ is negative ($x < 1$): In this case, the equation becomes: $-(x-1)-1=x-2$ Simplify: $-x+1-1=x-2$ $-x=x-2$ Now, let's isolate $x$: $-x-x=-2-1$ $-2x=-3$ $x=\frac{-3}{-2}$ $x=\frac{3}{2}$

So, for $x < 1$, the solution is $x = \frac{3}{2}$.

Now, we have two solution intervals:

• For $x < 1$: $x = \frac{3}{2}$
• For $x \geq 1$: Any value of $x$ in this range will satisfy the equation.

2. $2x-2|-|x| \geq x$

Let's solve this inequality by considering different cases based on the sign of $x$:

Case 1: $x \geq 0$: In this case, the inequality becomes: $2x-2-x \geq x$ Simplify: $x-2 \geq x$ This inequality is not possible for any value of $x$ because subtracting $2$ from $x$ makes it smaller, and it can never be greater than itself. Therefore, there is no solution in this case.

Case 2: $x < 0$: In this case, the inequality becomes: $2x-2-(-x) \geq x$ Simplify: $2x-2+x \geq x$ Combine like terms: $3x-2 \geq x$ Subtract $x$ from both sides: $2x-2 \geq 0$ Add $2$ to both sides: $2x \geq 2$ Divide both sides by $2$ (note that we can safely divide by $2$ because $2$ is positive): $x \geq 1$

So, for $x < 0$, the solution is $x \geq 1$.

In summary, the solution to the inequality $2x-2|-|x| \geq x$ is $x \geq 1$ when $x < 0$, and there is no solution when $x \geq 0$.