Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2,026 questions
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Proofs involving strict inequalities

let $a,b \in R$ Prove that if $3 \lt a \lt 5$ and $b= 2 + \sqrt{a-2}$ then, $3 \lt b \lt a$ My approach was simply to start with the first inequality and transform it into b and see what happens. ...
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Nonstandard inequality with parameter and absolute value

The given inequality is $|x^2-ax+1|<3(x^2+x+1)$. The question is: For which values of $a$, every $x$ is a solution? I am trying to solve it by making the graphics of the two sides of the ...
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Using triangle inequality to find $\lim_{(x,y)\to(0,0)}\frac{x^3-x^2y}{x^2+y^2+xy}$

This is an exercise from my textbook where the problem is to find the limit of the function $\frac{x^3-x^2y}{x^2+y^2+xy}$ when $(x,y) \to (0,0)$. So after changing to polar coordinates and ...
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An $\arcsin$ inequality

Show that if $0<|x|,|y|<1$, then $$\arcsin |x| +\arcsin |y| > \arcsin\left|\frac{x+y}{1+xy}\right|.$$ I found a proof (see below). Is there a different way (hopefully simpler) to show that ...
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Simplifying $\cos(2\arcsin(x))$ using only pythagorean trigonometric identity

I know that one can simplify $\cos(2\arcsin(x))$ using $\cos(a+b)=\cos(a)\cdot\cos(b)-\sin(a)\cdot\sin(b)$: \begin{alignat}{1} \cos(2\arcsin(x))&=\cos^2(\arcsin(x))-\sin^2(\arcsin(x)) \\&=1-2\...
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Absolute Value Rational Inequalities Help Please

I have been read plenty of questions on questions like this, but I still dont quite get it. For example, this question: $$\left| \frac{2x+1}{x-3} \right| \ge 2$$ How would I go about solving this?...
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Real Analysis introductory absolute value proof

let $x \in R$ Prove that $\vert x\vert \leq 2$ implies $\vert x^2 -4 \vert \leq 4 \vert x-2 \vert$ Here is my work: $\vert x-2 \vert \vert x + 2 \vert \leq 4 \vert x-2 \vert$ By the first ...
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Triangle Inequality and absolute value

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example: $$||a| - b| \le |a - b|$$ If $b \ge 0$, then $|b|$ is ...
Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral
I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: I=\...