Solve doesn't return all solutions
What is the difference between Reduce and Solve?
Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$. Prove that for every $a, b\in R\setminus\{0\}$ is correct this inequality:  $$\frac{|a+b|}{1+|a+b|}&lt;\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}.$$ Show lim sup (an) = lim inf (an) for increasing for increasing sequence of subsets. Version: Ubuntu 16.04  I have used it some weeks, but now it gives me this message when I'm trying to install software and update or upgrade:  Could not get lock /var/lib/apt/lists/lock - open (11: Resource temporarily unavailable) /var/lib/apt/lists/ locked   followed by this:    .  and another error:  ** (appstreamcli:4560): WARNING **: Exception: Unable to get write lock on /var/cache/app-info/xapian/default: already locked   What should I do? Increasing the size of tyres on a hybrid We are planning on doing some mountain biking in a few weeks time and we'd like to get some wider tyres for my wife's hybrid. I know there are similar questions and answers out there on this topic, but I need some specific help to make sure I buy the right tyres.  The bike is a Ladies Scott Sportster 50 2015, the best website I could find with a fairly accurate spec is     It's currently got Kenda K-180 700 x 38c / 30TPI tyres on it, which are fairly good all rounders and capable of running on clean forest paths but I don't think they would handle anything heavier.  It's got disk brakes so there isn't a clearance issue there, but I have noticed the frame is generally narrower than true MTBs, here are some photos i took of it.    Has anyone put larger, knobblier tyres on this kind of hybrid? whats the largest I can get away with? - I'm also slightly confused by all the tyre sizing, I'm sure someone told me this is a 27.5" tyre, but 700 seems to mean 29" on a lot of websites.  Any advice is very much appreciated, Thanks. elementary row operation change the eigenvalue of a matrix I know that elementary row operations may change the eigenvalues of a matrix, but if these operations don't change the determinant, why should change the eigenvalues??  Thanks Arithmetic progression topology Detect when an iOS app is launched for the first time? How do I detect when an iOS app is launched for the first time? Tufte advocates to "minimize the ink-to-data ratio" for charts. I am trying to style a multi data series bar chart, that can have a grid and axes as shown below (chart at the left is from the Economist and on the right my attempt).      Here is the code:  \documentclass[justified]{tufte-book} \usepackage{pgfplots,lipsum} \pgfplotsset{compat=newest} \begin{document} \section{Bar Charts}  \lipsum[1-2]\vfill   \begin{figure*}[htbp]   \fbox{  \begin{tikzpicture}   \centering   \begin{axis}[         ybar,           title={Cumulative Progress of Works},         height=8cm,         width=15.5cm,         bar width=0.4cm,         ymajorgrids, tick align=inside,         enlarge  y limits={value=.2,auto}, % see the upper          ymin=0,         ymax=100,         axis x line=bottom,          axis y line=right,         %enlarge x limits =e,         legend style={at={(0.5,-0.2)},         anchor=north,legend columns=-1},         ylabel={Percentage (\%)},         symbolic x coords={            Sep-11,Oct-11,Nov-11,Dec-11,            Jan-12,Feb-12,            Mar-12,           Apr-12},        xtick=data,        nodes near coords,        every node near coord/.append style={         anchor=mid west,         rotate=70       }     ]     \addplot coordinates {       (Sep-11,75.4064)       (Oct-11, 72.7961)        (Nov-11,94.4597)       (Dec-11,66.6786)        (Jan-12,67.5600)        (Feb-12,88.2339)       (Mar-12,78.6138)        (Apr-12,58.9129) };    \addplot coordinates {       (Sep-11,75.4064)       (Oct-11, 89.7961)        (Nov-11,94.4597)       (Dec-11,76.6786)        (Jan-12,77.5600)        (Feb-12,78.2339)       (Mar-12,88.6138)        (Apr-12,78.9129) };    \addplot coordinates {       (Sep-11,75.4064)       (Oct-11, 89.7961)        (Nov-11,94.4597)       (Dec-11,76.6786)        (Jan-12,77.5600)        (Feb-12,78.2339)       (Mar-12,88.6138)        (Apr-12,78.9129) };      \legend{First Fix,Second Fix,Third Fix}   \end{axis}   \end{tikzpicture} }  \caption{\protect\raggedright Cumulative progress for all MEP works. Notice the slower rate of production during the last three months.} \end{figure*}  \lipsum[1-3] \end{document}   I have moved the y-axis to the right, but cannot get rid of the axis line. The plot is also not fitting within the plot width, despite using \enlarge.   Edit  And thanks to Jake the ugly duckling has been transformed to the following, with minor variations to the code as suggested by Jake. I have only added a stronger color for one of the bars in order to highlight it: How to implement an abstract class in ruby? I know there is no concept of abstract class in ruby. But if at all it needs to be implemented, how to go about it? I tried something like...  class A   def self.new     raise 'Doh! You are trying to write Java in Ruby!'   end end  class B &lt; A   ...   ... end   But when I try to instantiate B, it is internally going to call A.new which is going to raise the exception.  Also, modules cannot be instantiated but they cannot be inherited too. making the new method private will also not work. Any pointers? Where are Time Machine local backups/snapshots stored? Out of curiosity, I'm wondering where in the file system Time Machine stores its local backups/snapshots. Does a US citizen traveling by plane in Mexico (international flight), for example from Mexicali to Mexico City need to have a passport? Where does Minecraft for Windows 10 store its data? I have searched in %appdata%, pulling up the list of files there, but there is no file labelled .minecraft. Searching %appdata%\.minecraft returns "no results found".   I also opened Minecraft to click on the resources button and could not locate that either. Changing units in QGIS: degrees to meters I have a global map with CRS WGS84-EPSG:4326 and I want to calculate the area of my polygons. From what I read, to change the unit from degrees to meters I need to change my CRS. However, there is no global system for this. Is there any solution to change the unit from degrees to meters for a global map? Is there any SE Site where Questions related to Hacking are on-topic? How to prevent Command-I/Command-Shift-I from opening Mac Mail when in browser? When in a browser (Chrome or Safari), the following keyboard shortcut will always bring up a window that says "Welcome To Mail":  Command+Shift+I  I would like to use this shortcut for another purpose.  Is there any way to disable this shortcut? 9 pirates have to divide 1000 coins... A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins.  Arriving on a deserted island, they now have to split up the loot. You, as the captain of the band, have to propose a distribution plan (who gets what). What's your proposal?  Consider that this bunch is a democratic lot. If your proposal is accepted by half of the group, then everybody adheres to it. However, if folks feel you are getting greedy, and less than half of the band agrees to your proposal, then they kill you, and then your First Mate gets to make a proposal. And so it goes in decreasing order of hierarchy/seniority. How to show that $\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3?$ Given the integral $(1)$     $$\int_{0}^{\pi}{\sin^5 x\over 1+\cos^3 x}\mathrm dx=\ln 3\tag1$$   This integral seems to be quite tough, I can't even know where to start. I have try substitution $u=1+\cos^3 x$ and $u=\cos^3 x$, can seem to remove the x variables.  We have the following identities   $$\sin^5a={5\over 8}\sin a-{5\over 16}\sin 3a+{1\over 16}\sin 5a\tag2$$  $$\cos^3 a={3\over 4}\cos a+{1\over 4}\cos 3a\tag3$$  We may write $(1)$ as  $$\int_{0}^{\pi}{\sin x(1-\cos^2 x)^2\over 1+\cos x(1-\sin ^2 x)}\mathrm dx\tag4$$  $u=1-\sin^2 x\implies \mathrm du=-2\sin x\cos x\mathrm dx$ then $(4)$ becomes  $${1\over 2}\int_{0}^{1}{(1-u)^2\over \sqrt{u}+u^2}\mathrm du\tag5$$  $u=v^2\implies du=2vdv$ then $(5)$ becomes  $$\int_{0}^{1}{(1-v^2)^2\over v+v^4}\cdot{v\mathrm dv}\tag6$$ Simplify to  $$\int_{0}^{1}{(1-v)(1-v^2)\over 1-v+v^2}\mathrm dv\tag7$$  I guess using partial decomposition of fraction and integrate term by term, seem lengthy process  How else can we prove $(1)$? Are there any tools for Windows 7 that will allow me to enable/disable monitors from the command-line?  For example, my GTX 480 has three outputs (2x DVI and 1x HDMI) but only two displays can be active at any given time.  I'd like for a command-line to allow me to disable one of the DVI monitors and enable the HDMI monitor, etc.  (I've found NirCmd which is an amazing utility but it doesn't seem to support this...)  EDIT: UltraMon seems capable of doing this but I really hoping for a more light-weight command-line solution... Anything similar in javascript to ruby's #{value} (string interpolation) Why do cockroaches burst from soapy water?