Řešení rovnice s absolutními hodnotami
Řeš v dané množině rovnici:
\( \left| {2 x + 6} \right|\ -\ \left| {4\ –\ 2 x} \right| + \left| {x + 7} \right| = 4 \) v \( \langle-2;\ 2) \)
\( I. x \in \left. {\langle – 2;\ 2} \right) \)
\( \left| {2 x + 6} \right|\ –\ \left| {4\ –\ 2 x} \right| + \left| {x + 7} \right| = 4 \)
\( 2 x + 6\ –\ \left( {4\ –\ 2 x} \right) + x + 7 = 4 \)
\( 2 x + 6\ –\ 4 + 2 x + x + 7 = 4 \)
\( 5 x =\ 0\ \ \ \ \ /:5 \)
\( x =\ 0 \)
\( K=\left\{{ 0}\right\} \)<
\( I. x \in \left. {\langle – 2;\ 2} \right) \)
\( \left| {2 x + 6} \right|\ –\ \left| {4\ –\ 2 x} \right| + \left| {x + 7} \right| = 4 \)
\( 2 x + 6\ –\ \left( {4\ –\ 2 x} \right) + x + 7 = 4 \)
\( 2 x + 6\ –\ 4 + 2 x + x + 7 = 4 \)
\( 5 x =\ 5\ \ \ \ \ /:5 \)
\( x =\ 1 \)
\( K=\left\{{ 1}\right\} \)<
\( I. x \in \left. {\langle – 2;\ 2} \right) \)
\( \left| {2 x + 6} \right|\ –\ \left| {4\ –\ 2 x} \right| + \left| {x + 7} \right| = 4 \)
\( 2 x + 6\ –\ \left( {4\ –\ 2 x} \right) + x + 7 = 4 \)
\( 2 x + 6\ –\ 4 + 2 x + x + 7 = 4 \)
\( 5 x =\ – 5\ \ \ \ \ /:5 \)
\( x =\ – 1 \)
\( K=\left\{{ – 1}\right\} \)
\( I. x \in \left. {\langle – 2;\ 2} \right) \)
\( \left| {2 x + 6} \right|\ –\ \left| {4\ –\ 2 x} \right| + \left| {x + 7} \right| = 4 \)
\( 2 x + 6\ –\ \left( {4\ –\ 2 x} \right) + x + 7 = 4 \)
\( 2 x + 6\ –\ 4 + 2 x + x + 7 = 4 \)
\( 5 x =\ – 5\ \ \ \ \ /:5 \)
\( x =\ 1 \)
\( K=\left\{{ 1}\right\} \)