【数学I】例題1.3.7：絶対値記号を含む方程式・不等式1（One More）★★

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% 例題I1.3.7：絶対値記号を含む方程式・不等式1（One More）★★
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次の方程式，不等式を解け．

(1) $|x-3|=2$

(2) $|x-1|\leqq 5$

(3) $|x-4|>3$

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% 例題I1.3.7の解答（One More）★★
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(1) $|x-3|=2$より，$x-3=\pm 2$

よって，$x=1,5$

(2) $|x-1|\leqq 5$より，$-5\leqq x-1\leqq 5$

よって，$-4\leqq x\leqq 6$

(3) $|x-4|>3$より，$x-4<-3,3<x-4$

よって，$x<1,7<x$

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% 問題I1.3.7：絶対値記号を含む方程式・不等式1（One More）★★
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次の方程式，不等式を解け．

(1) $|x+2|=3$

(2) $|x-5|\leqq 4$

(3) $|x+1|>2$

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% 問題I1.3.7の解答（One More）★★
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(1) $|x+2|=3$より，$x+2=\pm 3$

よって，$x=1,-5$

(2) $|x-5|\leqq 4$より，$-4\leqq x-5\leqq 4$

よって，$1\leqq x\leqq 9$

(3) $|x+1|>2$より，$x+1<-2,2<x+1$

よって，$x<-3,x>1$

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% 例題I1.3.7の側注：絶対値記号を含む方程式・不等式1（One More）★★
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% 側注1
$x-3=X$とおくと，$|X|=2$

よって，$X=\pm 2$

% 側注2
$x-1=X$とおくと，$|X|\leqq 5$

よって，$-5\leqq X\leqq 5$

% 側注3
$x-4=X$とおくと，$|X|>3$

よって，$X<-3,3<X$

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% 問題I1.3.7の側注：絶対値記号を含む方程式・不等式1（One More）★★
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% 側注1
$x+2=X$とおくと，$|X|=3$

よって，$X=\pm 3$

% 側注2
$x-5=X$とおくと，$|X|\leqq 4$

よって，$-4\leqq X\leqq 4$

% 側注3
$x+1=X$とおくと，$|X|>2$

よって，$X<-2,2<X$

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% 例題I1.3.7の考え方・補足など：絶対値記号を含む方程式・不等式1（One More）★★
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% 考え方
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絶対値記号を含む方程式・不等式を解くには，場合分けをして絶対値記号を外すことが基本である．

ただし，(1)〜(3)のような右辺がすべて正の定数の場合は，$|x|$を数直線上の原点Oから任意の点$x$までの距離と考えることができるので，次の性質を用いて解くとよい．

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% 絶対値と方程式・不等式
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$a>0$のとき，

\begin{align*}
({\mathrm{i}})&|x|=a\text{の解は，}x=\pm a\\
({\mathrm{ii}})&|x|<a\text{の解は，}-a<x<a\\
({\mathrm{iii}})&|x|>a\text{の解は，}x<-a,x>a
\end{align*}

(i)

(ii)

(iii)