【数学I】例題1.3.4：不等式を満たす整数の解（One More）★★

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% 例題I1.3.4：不等式を満たす整数の解（One More）★★
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(1) 不等式$4x-2<2x+4$を満たす自然数$x$の値をすべて求めよ．

(2) 不等式$4x+a<5x<3x+10$を満たす整数$x$がちょうど2個存在するような定数$a$の値の範囲を求めよ．

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% 例題I1.3.4の解答（One More）★★
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(1) 与えられた不等式より，$2x<6$

したがって，$x<3$

よって，$x$は自然数であるから，$x=1,2$

(2) $4x+a<5x$を解くと，$-x<-a$より，$x>a\cdots(\mathrm{i})$

$5x<3x+10$を解くと，$2x<10$より，$x<5\cdots(\mathrm{ii})$

(i)，(ii)より，不等式を満たす整数$x$がちょうど2個となるのは右の図のような場合である．

よって，$2\leqq a<3$

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% 問題I1.3.4：不等式を満たす整数の解（One More）★★
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(1) 不等式$3x-1<2x+5$を満たす自然数$x$の値をすべて求めよ．

(2) 次の連立不等式を満たす整数$x$がちょうど2個存在するような定数$a$の値の範囲を求めよ．

\[
\{\begin{array}{l}
3x+a<4x\\
3x>4x-7
\end{array}
\]

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% 問題I1.3.4の解答（One More）★★
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(1) 与えられた不等式より，$x<6$

したがって，$x$は自然数であるから，$x=1,2,3,4,5$

(2) $3x+a<4x$を解くと，$-x<-a$より，$x>a\cdots(\mathrm{i})$

$3x>4x-7$を解くと，$-x>-7$より，$x<7\cdots(\mathrm{ii})$

(i)，(ii)より，不等式を満たす整数$x$がちょうど2個となるのは右の図のような場合である．

よって，$4\leqq a<5$

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% 例題I1.3.4の側注：不等式を満たす整数の解（One More）★★
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% 側注1
数直線を用いるとよい．

% 側注2
等号を含むか否かに注意すること．

% 側注3
$3$は含まない．

% 側注4
$A<B<C$より，

\[
\{\begin{array}{l}A<B\\B<C\end{array}
\]

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% 問題I1.3.4の側注：不等式を満たす整数の解（One More）★★
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% 側注1
数直線を用いるとよい．

% 側注2
等号を含むか否かに注意すること．

% 側注3
$6$は含まない．

% 側注4
$A<B<C$より，

\[
\{\begin{array}{l}A<B\\B<C\end{array}
\]

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% 例題I1.3.4の考え方・補足など：不等式を満たす整数の解（One More）★★
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(1) 不等式を解き，その解の中から条件に適する自然数を考える．このとき，与えられた不等式は等号を含まないので注意すること．

(2) 不等式を解き，その解を満たす整数$x$がちょうど2個存在する定数$a$の範囲を，数直線を用いて考える．このとき，等号を含むか否かをよく考えて注意すること．

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等号を含むか否かをよく考える．

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(2)は$a=2,3$のときに適するか否かを考えて注意すること．$a=2$のとき

$2<a<3$のとき

$a=3$のとき上の図より，$a=2,2<a<3$のとき，$x=3,4$となるので適することがわかる．$a=3$のとき，$x=4$となるので不適である．