Is it true that $|x + y| \geq |x| - |y|$?
Help with proof of $|x+y| \geq |x|-|y|$
Do synonyms have exactly the same meanings as each other in all contexts? I have a question about synonyms: I'm wondering for quite a while if synonyms always stand exactly for the same thing. Is there sometimes a little difference in meaning? Let's use clumsy and clunky for example. Can I always replace clunky with clumsy? Since I'm not a native speaker, as you may have noticed, I cannot tell them apart. According to my dictionary, they mean exactly the same. Is it always true?  If the question has already been ask, sorry I couldn't find it.  Edit: Thank you very much! Every answer has helped a lot and thank you so much for it. I really appreciate it. I've voted all your answer because every answer helped me in a different way. I've accepted Barrie England's answer because it was the best one in my case. But all of yours were really good and helpful :) Are there any differences between Skyrim & Skyrim: Special Edition? No more updates for Google Chrome 32-bit I am using Google Chrome 48.0 32-bit. Today when I launched Google Chrome, I got this message:    This computer will soon stop receiving Google Chrome updates because this Linux system will no longer be supported   I thought Ubuntu 14.04 is supported for five years, what's exactly happening here?  And on 64-bit Ubuntu, I get an error from Update Manager and apt-get update:  Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release  Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)   Some index files failed to download.   They have been ignored, or old ones used instead. Below is my code to create the component dynamically - The component is "ui:inputselect"  $A.createComponent(             RecoveryLineItem.Attributes[i].name,{                 "class" : "slds-input ",                 "aura:id" : 'ExampleId',                 "label" : RecoveryLineItem.Attributes[i].label,                 "multiple" : RecoveryLineItem.Attributes[i].multiple             }, ....   ..  console.log(component.find('ExampleId').get("v.value"))    Or console.log(component.find('ExampleId'))  is Giving as 'Undefined"  How to access the aura:id of the dynamically created components? How can I ensure my trip to the UK will not have to be cancelled because of Brexit? I've got a trip planned to the UK in June. Due to brexit, I'm worried what changes may be introduced before my arrival.  I'm traveling on a EU / Polish passport. I'm going for a couple weeks of holiday.  Is there anything I can do right now to make it less likely any changes won't stop me at the immigration? For example, applying for some tourist visa which does not rely on the EU status? Who paid for Harry's Nimbus 2000? The Nimbus 2000 was the fastest broom at the time of the Harry's first year at Hogwarts.  Professor McGonagall wanted to win the Quidditch Cup so she spoke with Dumbledore and got an exception from the rule that no first-years could have a broom for Harry.  My question is who paid for the broom?  It couldn't have been a cheap broom.  They wouldn't have taken gold out of Harry's Gringotts account without permission.  It wouldn't be right for the school to pay for it and I doubt McGonagall paid for it out of her own pocket.  So who paid for the broom? What does "it" refer to in "it's raining"? I wanted to leave the question title as is so as not to take away from my amusement :).  Anyway,     It's raining.   What is raining? Is it the sky? The clouds? The weather? The rain? What is "it"? Any historical insights on the statement? QGIS area calculation differs when on the fly CRS transformation enabled When I open QGIS, add the layer, and calculate the areas of the shapefile via field calculator I get a different area than when I open QGIS and check "Enable on the fly CRS transformation" and calculate the area. This is despite making sure that the project and the layer have the same Coordinate system (same EPSG number). What am I doing wrong?   I have a shapefile with area calculations made with ArcGIS (not be me, the data was handed to me and I have no clue for which CRS the area was calculated with ArcGIS). The shapefile layer CRS is EPSG: 21781 (Switzerland). In QGIS, if I do not change the OTF settings and leave the project CRS as EPSG:4326 (WGS84) I get the same value as the ArcGIS area value. However, if I change the OTF before adding the layer to EPSG: 21781 I get different area values. As I understand this suggests that ArcGIS Area was calculated with the CRS EPSG: 4326.  First workflow:   open QGIS  project CRS: EPSG 4326  add layer  project CRS adapts automatically and is EPSG 21781 now  calculate $area with field calculator    Second workflow:   open QGIS  project CRS: EPSG 4326  Turn on OTF, set project CRS to EPSG 21781  add layer  calculate $area with field calculator    Step 5 of the first and second workflow DO NOT produce the same area. Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$  So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $a_{n}$ and the denominator be $b_{n}$ Is there a way to use this statement so that I could force the original sequence into the form of $1/\left(1+\frac{1}{n}\right)^n$ stl::multimap - how do i get groups of data? Multimap essentially has groups of data sorted by the key. I want a method by which I could access these individual groups and get their aggregate values. For example, in a std::multimap&lt; string, int &gt; I store   {"Group1", 1},  {"Group1", 2},  {"Group1", 3},   {"Group2", 10},  {"Group2", 11},  {"Group2", 12}   Having stored these values, I should be able to iterate this multimap and get the aggregate values of each "group". Problem is there aren't any functions defined in STL to access MultiMaps in such a way. I could use lower_bound, upper_bound to manually iterate the multimap and total the group's contents, but I am hoping there could be better ways already defined in STL ? Can anyone propose a solution as to how I could get the aggregate values for a group in the above example. Is there a difference in disenchanting strong items vs. weak ones? Possible Duplicate:           states that the strength of the enchantment doesn't matter when disenchanting items to learn new enchantments.  Am I reading that right? If I have novice robes with 50% faster regen and disenchant it and my enchantment skill is at 5/5, when I enchant that item will it be the same result as disenchanting master robes with 150% faster regen? Because if you do the math, 5/5 enchanting will give you 100% better enchantment.   Therefore wouldn't it be better to disenchant more powerful items than weaker ones? Is it safe to answer "erase disk and install Ubuntu" on a virtual machine? LOAD DATA INFILE easily convert YYYYMMDD to YYYY-MM-DD? How to calculate time difference in java? Installing Cygwin packages from the command-line Possible Duplicate:        I'm looking for a "yum"-link method to install additional Cygwin packages from the command-line. Let's say I want to add the package "Math/bc", then ideally I'd like to be able to do it  using one command as in:  yum install bc   Currently the only way I've found to install new packages is to locate and run the installation file SETUP.EXE file and then scroll to the relevant package, add it, etc. This is  quite cumbersome so I'm sure there is a smarter command-line way to add an additional package. When, where and **how often** do you find polynomials of higher degrees than two in mathematical, pure/applied, research? A formula for solving a polynomial of degree three, see  link; $ax^3+bx^2+cx+d=0$, is  $$\begin{align} x\quad&amp;=\quad       \sqrt[3]{           \left(               \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}           \right)            +                 \sqrt{               \left(                   \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}               \right) ^2                 +                \left(                   \frac{c}{3a} - \frac{b^2}{9a^2}               \right) ^3            }       }\\       &amp;+\quad       \sqrt[3]{           \left(               \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}           \right)            -                 \sqrt{               \left(                   \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}               \right) ^2                 +                \left(                   \frac{c}{3a} - \frac{b^2}{9a^2}               \right) ^3            }       }       \;-\;\frac{b}{3a} \end{align}$$     Unlike quadratic, cubic, and quartic polynomials, the general quintic   cannot be solved algebraically in terms of a finite number of   additions, subtractions, multiplications, divisions, and root   extractions, as rigorously demonstrated by Abel (Abel's impossibility   theorem) and Galois. However, certain classes of quintic equations can   be solved [...] Source:    At levels of $5^{\text{th}}$ degree polynomials, things are starting to look really serious in my eyes. My question is:     If it is possible to not answer subjectively: When, where and how often do you find polynomials of higher degrees than two in mathematical, pure/applied, research? Tweaking AStar to find closest location to unreachable destination I've implemented AStar in Java and it works ok for an area with obstacles where the chosen destination is reachable.  However, when the destination is unreachable, the calculated "path" is in no way to the closest location (to the unreachable location) but is instead some random path.  Is there a feasible way to tweak AStar into finding the path to the closest location to an unreachable destination? How to make tr aware of non-ascii(unicode) characters? What flash slaves can be used with a Canon Speedlite 600EX-RT? My current flash setup is a standard shoe mount but I've been thinking about getting a remote.  I've been drooling over the Speedlite 600EX-RT. It uses radio to control slaves, which is nice (longer distance, doesn't require line of sight, etc.).  One thing I don't understand: do slaves also have to be 600EX-RTs or do other flash units use radio to communicate already? Can any (cheap?) flash slaves can be used with a Canon Speedlite 600EX-RT or do I need to buy the 600EX and a ST-E3-RT transmitter in order (or multiple 600EX-RTs) to have a radio-controlled remote flash setup? Can we have a notification when the answer to any of my question got edited? If a user asked a question and got an answer which was not helpful.  After sometime the answerer edits the answer with some good content. How the OP will get to know that?  Is it necessary to add comment check my edited answer?  We can say that the OP should refresh the page to find the helpful answer.  Consider if the OP finds the answer as helpful and accepted that. After sometime if the answerer/anyother user edit the answer to change the content which may not solve the issue.   How will it be notified to OP?  currently we are notifying the OP when there is any comment to the queation or any new answers or any comments with the name of OP with @ to any answer.  This will lead the future visitors to confuse and reduce the quality of post.  Can we have a notification to the original poster when the answer to any of his/her question got edited?