# Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2,010 questions
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### Absolute Value Problem Using Only Variables

I recently encountered this problem, and it does not make sense to me. It looks like Given $|a(b-cx)|=d$ , find the value of $|x-\frac{b}{c}|$ This was on a multiple choice test awhile ago, and I ...
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### For what range of a-values does $||x|-2|=a$ have only 2 possible answers? (without drawing a graph)

How can i solve "For what range of a-values does the equation $||x|-2|=a$ have only 2 possible answers?" Without using graphs by using a graph I know the answer is a>2 or {0} as in the picture :
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### Is $y=|x^3|$a parabola?

I'm just curious. It seems to have the same shape and a similar form as parabolas such as $x^2$ and $x^4$. The odd exponent would normally give negative outputs for negative inputs, but the absolute ...
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### Understanding the absolute value of a matrix.

I believe that the absolute value of a matrix is defined as $$|A|=\sqrt{A^{\dagger}A} \ .$$ But the square root of a matrix is not unique wikipedia gives a list of examples to illustrate this. To ...
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### Solve for $x\in\mathbb{R}$: $x^{2} + 2|x-3| - 10 \leq 0$

I went about taking cases for $x^{2} + 2|x-3|- 10\leq 0$. Taking $x-3$ and $-(x-3)$ as cases. Is it the correct approach? Taking cases I realise I get $x^{2} + 2x - 16 \leq 0$ and $x^{2}-2x-4 \leq 0$...
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### Help me understand $y=f(x)$ vs. $y=f(|x|)$ intuitively

I am writing this because I want to fill the hole in my understanding of elementary functions in Euclidean plane. In class, we discussed parallel examples, such as $y = f(x)$ vs. $y = |f(x)|$. ...
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### Inequation with absolute values

How would one proceed in solving this difficult inequation with multiple absolute values? is there a way one should proceed ? $$\frac{x}{||x|-2|} \le \frac{x-1}{|x-3|}$$ thanks
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### The Median Minimizes the Sum of Absolute Deviations (The ${L}_{1}$ Norm)

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x|$$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and ...
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### Redefinition of a Constant Leading to Nullification of Absolute Value

I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
### bound for $a,b,c$ in $|ax^2+bx+c| \leq 1\;\forall x\in \left[0,1\right]$
Consider $a,b,c\in\mathbb{R}$ such that $|ax^2+bx+c|\leq 1\;\forall x\in \left[0,1\right]$. Prove that $|a|\leq 8\;\;,|b| \leq 8$ and $|c| \leq 1$. My Attempt: Set $x = 0$ in $|ax^2+bx+c|\leq 1$ to ...
Given the equation $|x+2| + |3x+6| = 8,$ how can I find the sum of all its roots?