【数学I】例題1.3.2：1次不等式（基本）（One More）★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.2：1次不等式（基本）（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の1次不等式を解け．

(1) $3x-2<4x+1$

(2) $2(x+1)\leqq 3(2x-1)$

(3) $\frac{x+1}{6}-4\geqq\frac{x}{3}-\frac{7}{2}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.2の解答（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 移項すると，$3x-4x<1+2$整理すると，$-x<3$

よって，$x>-3$

(2) 展開すると，$2x+2\leqq 6x-3$移項すると，$2x-6x\leqq-3-2$

整理すると，$-4x\leqq-5$

よって，$x\geqq\frac{5}{4}$

(3) 両辺に$6$を掛けると，$x+1-24\geqq 2x-21$移項すると，$x-2x\geqq-21+23$

整理すると，$-x\geqq 2$

よって，$x\leqq-2$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.3.2：1次不等式（基本）（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の1次不等式を解け．

(1) $5x+1<3x-4$

(2) $4(2x-3)>3(x+2)$

(3) $\frac{2x+3}{4}-\frac{x-1}{6}\geqq\frac{1}{3}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.3.2の解答（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 移項すると，$5x-3x<-4-1$整理すると，$2x<-5$

よって，$x<-\frac{5}{2}$

(2) 展開すると，$8x-12>3x+6$移項すると，$8x-3x>6+12$

整理すると，$5x>18$

よって，$x>\frac{18}{5}$

(3) 両辺に$12$を掛けると，$3(2x+3)-2(x-1)\geqq 4$展開すると，$6x+9-2x+2\geqq 4$

整理すると，$4x+11\geqq 4$

移項すると，$4x\geqq-7$

よって，$x\geqq-\frac{7}{4}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.2の側注：1次不等式（基本）（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
不等式の両辺に$-1$を掛ける，もしくは両辺を$-1$で割ると考える．このとき，不等号の向きが変わるので注意すること．

% 側注2
不等号の向きが変わる．

% 側注3
不等号の向きが変わる．

% 側注4
移項すると符号が変わる．

% 側注5
不等式の両辺に，$2,3,6$の最小公倍数$6$を掛ける．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.3.2の側注：1次不等式（基本）（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
移項すると符号が変わる．

% 側注2
不等式の両辺に，$4,6,3$の最小公倍数$12$を掛ける．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.2の考え方・補足など：1次不等式（基本）（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 考え方
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$ax>b$または$ax<b$の形や，$ax\geqq b$または$ax\leqq b$の形に変形する．また，両辺に負の数を掛ける（負の数で割る）ときは，不等号の向きが変わるので注意すること．

(3) 両辺の分数の形に着目し，分母の最小公倍数を掛けてから不等式の解を求めるとよい．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% One Point
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

不等式の両辺に，負の数を掛ける（負の数で割る）ときは，不等号の向きが変わる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 注意
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

本書では，数直線上に不等式の解を示すとき，「$>,<$」の場合は$\circ$，「$\geqq,\leqq$」の場合は$\bullet$を用いて表す．

$x>a$

$x\geqq a$