# Tagged Questions

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### Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
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### Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
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### Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
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### Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
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### How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
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### $\left | -(x+2)^2+6(x+2) \right |>13$

I did $-(x+2)^2+6(x+2)>13$ and $-(x+2)^2+6(x+2)< -13$. The first inequality had complex solutions and therefore can be disregarded but the second one has two real solutions, $x \approx -3.7$ and ...
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### Absolute value quadratic inequalities not the usual?

$| -x^2 + 6x | \gt 13$,for example. I would start off solving $-x^2 + 6x = \pm 13$ and either get 4 solutions, 3 solutions or two simply do the the nature of the graph. Without knowing if the two ...
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### How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$? [duplicate]

Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$ I think this problem may be solved using nice methods, but I can't find ...
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### Prove that: $\left | 2x-y-4 \right |\geq 4\sqrt{2}+4$

Let $x,y\in \mathbb{R}$ know that $4x^2-9y^2=36$ Prove that: $$\left | 2x-y-4 \right |\geq 4\sqrt{2}+4$$
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### Finding the number of integral values of a for which the inequality $3- |{x-a}|>x^2$ is satisfied by at least one negative value of $x$.

Finding the number of integral values of a for which the inequality$$3- |{x-a}|>x^2$$ is satisfied by at least one negative value of $x$.Here's a duplicate post ->{{Determine all the values of ...
### Questions about $|f(1+a+bi)|<|f(1+a)|$
Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$. Some questions : $1)$ If ...