問題の解答
検索用コード（LaTeX）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.1.12：絶対値記号を含む関数のグラフ3（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
不等式$|2x+2|+|x-1|>-x+2$をグラフを利用して解け．
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.1.12の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$y=|2x+2|+|x-1|$とする．
(i) $x<-1$のとき
\[
y=-(2x+2)-(x-1)=-3x-1
\]
(ii) $-1\leqq x<1$のとき
\[
y=(2x+2)-(x-1)=x+3
\]
(iii) $1\leqq x$のとき
\[
y=(2x+2)+(x-1)=3x+1
\]
したがって，(i)〜(iii)より，関数$y=|2x+2|+|x-1|$のグラフは右の図の（ア）となる．一方，関数$y=-x+2$のグラフは右の図の（イ）となる．右の図より，（ア）と（イ）のグラフは，$x<-1$または$-1\leqq x<1$の範囲で交わる．
（ア）と（イ）のグラフの交点の$x$座標は，$x<-1$のとき，$-3x-1=-x+2$より，$x=-\frac{3}{2}$
$-1\leqq x<1$のとき，$x+3=-x+2$より，$x=-\frac{1}{2}$
よって，不等式$|2x+2|+|x-1|>-x+2$の解は，
\[
x<-\frac{3}{2},-\frac{1}{2}<x
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.1.12：絶対値記号を含む関数のグラフ3（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
不等式$|x+4|+|2x-1|\leqq-2x+2$をグラフを利用して解け．
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.1.12の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$y=|x+4|+|2x-1|$とおく．
(i) $x<-4$のとき
\[
y=-(x+4)-(2x-1)=-3x-3
\]
(ii) $-4\leqq x<\frac{1}{2}$のとき
\[
y=(x+4)-(2x-1)=-x+5
\]
(iii) $\frac{1}{2}\leqq x$のとき
\[
y=(x+4)+(2x-1)=3x+3
\]
したがって，(i)〜(iii)より，関数$y=|x+4|+|2x-1|$のグラフは右の図の（ア）となる．一方，関数$y=-2x+2$のグラフは右の図の（イ）となる．右の図より，（ア）と（イ）のグラフは，$x<-4$または$-4\leqq x<\frac{1}{2}$の範囲で交わる．
（ア）と（イ）のグラフの交点の$x$座標は，$x<-4$のとき，$-3x-3=-2x+2$より，$x=-5$
$-4\leqq x<\frac{1}{2}$のとき，$-x+5=-2x+2$より，$x=-3$
よって，不等式$|x+4|+|2x-1|\leqq-2x+2$の解は，
\[
-5\leqq x\leqq-3
\]
あわせて読みたい
【数学I】3章：2次関数（基本事項）
検索用コード（LaTeX） 本文・解答側注 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 基本事項I3.1.1：関数（One More） %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%...
あわせて読みたい
【数学I】3章：2次関数（節末問題・章末問題）
節末I3.1.1〜I3.1.5の解答 節末I3.1.1節末I3.1.2節末I3.1.3節末I3.1.4節末I3.1.5 リンク（関連例題） https://onemath.net/onemorei-reidai3-1-1 https://onemath.net/o...