【数学I】例題1.3.6：文字を含む1次不等式（One More）★★★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.6：文字を含む1次不等式（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$a$を定数とするとき，次の問いに答えよ．

(1) $x$の不等式$ax-3<0$を解け．

(2) $x$の不等式$(a+1)x<a^2+a$を解け．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.6の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) $ax-3<0$より，$ax<3$

(i) $a>0$のとき，$x<\frac{3}{a}$

(ii) $a=0$のとき，不等式は，$0\cdot x<3$

したがって，解はすべての実数

(iii) $a<0$のとき，不等式は，$x>\frac{3}{a}$

よって，(i)〜(iii)より，求める解は，

\[
\{
\begin{array}{lll}
a>0&\text{のとき，}&x<\frac{3}{a}\\
a=0&\text{のとき，}&\text{すべての実数}\\
a<0&\text{のとき，}&x>\frac{3}{a}
\end{array}
\]

(2) $(a+1)x<a^2+a$より，$(a+1)x<a(a+1)$

(i) $a+1>0$，すなわち，$a>-1$のとき，$x<a$

(ii) $a+1=0$，すなわち，$a=-1$のとき，$0\cdot x<0$

これを満たす$x$の値はない．したがって，解なし

(iii) $a+1<0$，すなわち，$a<-1$のとき，$x>a$

よって，(i)〜(iii)より，求める解は，

\[
\{
\begin{array}{lll}
a>-1&\text{のとき，}&x<a\\
a=-1&\text{のとき，}&\text{解なし}\\
a<-1&\text{のとき，}&x>a
\end{array}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.3.6：文字を含む1次不等式（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$a$を定数とするとき，次の問いに答えよ．

(1) $x$の不等式$ax+2>0$を解け．

(2) $x$の不等式$(a-1)x\leqq a^2-a$を解け．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.3.6の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) $ax+2>0$より，$ax>-2$

(i) $a>0$のとき，$x>-\frac{2}{a}$

(ii) $a=0$のとき，不等式は$0\cdot x>-2$

したがって，解はすべての実数

(iii) $a<0$のとき，不等式は$x<-\frac{2}{a}$

よって，(i)〜(iii)より，求める解は，

\[
\{
\begin{array}{lll}
a>0&\text{のとき，}&x>-\frac{2}{a}\\
a=0&\text{のとき，}&\text{すべての実数}\\
a<0&\text{のとき，}&x<-\frac{2}{a}
\end{array}
\]

(2) $(a-1)x\leqq a^2-a$より，$(a-1)x\leqq a(a-1)$

(i) $a-1>0$すなわち，$a>1$のとき，$x\leqq a$

(ii) $a-1=0$すなわち，$a=1$のとき，$0\cdot x\leqq 0$

これを満たす$x$の値はすべての実数．したがって，解はすべての実数

(iii) $a-1<0$すなわち，$a<1$のとき，$x\geqq a$

よって，(i)〜(iii)より，求める解は，

\[
\{
\begin{array}{lll}
a>1&\text{のとき，}&x\leqq a\\
a=1&\text{のとき，}&\text{すべての実数}\\
a<1&\text{のとき，}&x\geqq a
\end{array}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.6の側注：文字を含む1次不等式（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$x$がどのような値でも，$0<3$であるので，すべての実数$x$について成り立つ．

% 側注2
$a<0$より，負の数$a$で割るので，不等号の向きが変わる．

% 側注3
$0<0$が成り立つような$x$の値はない．

% 側注4
$a<-1$より，負の数$a+1$で割るので，不等号の向きが変わる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I1.3.6の側注：文字を含む1次不等式（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$x$がどのような値でも，$0>-2$が成り立つので，すべての実数$x$について成り立つ．

% 側注2
$a<0$より，負の数$a$で割るので，不等号の向きが変わる．

% 側注3
$0\leqq 0$はすべての$x$で成り立つ．

% 側注4
$a<1$より，負の数$a-1$で割るので，不等号の向きが変わる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I1.3.6の考え方・補足など：文字を含む1次不等式（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

(1) $a\neq 0$であれば与えられた不等式は1次不等式となるが，$a=0$のときも含まれていることに注意すること．$a=0$のときは，両辺を$a$で割ることはできない（$0$で割ることはできない）．一般に，1次不等式$Ax>B$の解は，$A=0$のときは次のようになる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1次不等式$Ax>B$の解
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$A=0$のとき，不等式は，$0\cdot x>B$

よって，$B\geqq 0$のとき，解なしである．また，$B<0$のとき，解はすべての実数となる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 補足
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(2)のような，$Ax<B$の解も同様に考えられる．なお，負の数で割るとき，不等号の向きが変わることにも注意すること．