Άθροισμα απολύτων (που ανάγονται σε ένα) ίσο με απόλυτο
Λύση
![\[\begin{cases} \alpha - 2 = 2\alpha - 7 \\ \text{ή}\\ \alpha - 2 = -2\alpha + 7 } \end{cases}\Rightarrow\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-09256fc87d5c328af3995b5c48903933_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{cases} \alpha - 2\alpha = 2 - 7 \\ \text{ή}\\ \alpha +2\alpha = 2 + 7 } \end{cases} \Rightarrow\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-0f71ae5108d96798c1a3f1b6ac1cd091_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{cases} - \alpha = -5 \\ \text{ή}\\ \alpha +2\alpha = +2 + 7 } \end{cases} \Rightarrow\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-2009d58f32635d86d28018b6456c6822_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{cases} \alpha = 5 \\ \text{ή}\\ 3\alpha = 9 } \end{cases} \Rightarrow\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-840906aff593d1ffa7ded908d35a9fe3_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{cases} \alpha = 5 \\ \text{ή}\\ \alpha = 3 } \end{cases}\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-af67b66aa84219da2f32de0f561c30ea_l3.png "Rendered by QuickLaTeX.com")
ΑΚΟΛΟΥΘΕΙ ΑΝΑΛΥΤΙΚΗ ΕΠΕΞΗΓΗΣΗ ΤΗΣ ΛΥΣΗΣ
Η αρχική εξίσωση είναι:
![\[ |\alpha - 2| + |2 - \alpha| = |4\alpha - 14|. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-532eebeaf96a0d4ebe54a3d8138729ae_l3.png "Rendered by QuickLaTeX.com")
Γνωρίζουμε ότι 
, επομένως:
![\[ {\color{blue}{|\alpha - 2| + |2 - \alpha| = 2|\alpha - 2|. }}\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-87ace7fb6aced1c5743bce0e1d1e3c50_l3.png "Rendered by QuickLaTeX.com")
Άρα, η εξίσωση γίνεται:
![\[ 2|\alpha - 2| = |4\alpha - 14|. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-ae9df8b7bcd7703e7a043575c5b29890_l3.png "Rendered by QuickLaTeX.com")
Στη συνέχεια, παρατηρούμε ότι το δεξί μέλος της εξίσωσης είναι:
![\[ {\color{blue}{|4\alpha - 14| = |2(2\alpha - 7)| = 2|2\alpha - 7|. }}\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-dae3cb742d0f84056c3c3f8f61cb0bb0_l3.png "Rendered by QuickLaTeX.com")
Έτσι, η εξίσωση γίνεται:
![\[ 2|\alpha - 2| = 2|2\alpha - 7|. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-04fc2206d9f96b35fb62b09d4e85b50d_l3.png "Rendered by QuickLaTeX.com")
Απλοποιούμε διαίρεση με το 2:
![\[ |\alpha - 2| = |2\alpha - 7|. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-ca8ab8dd25569aa6908108f9a49e39da_l3.png "Rendered by QuickLaTeX.com")
Λύση της εξίσωσης
δύο απόλυτες τιμές είναι ίσες όταν οι παραστάσεις εντός των απόλυτων τιμών είναι ίσες ή αντίθετες ιδιότητα 10.δηλαδή:
![\[{\color{blue}{|\alpha| = | \beta | \Leftrightarrow \begin{cases} \alpha =\beta \\ \text{ή}\\ \alpha = - \beta } \end{cases}\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-ca791540bb0a5988b8335f7357734fa6_l3.png "Rendered by QuickLaTeX.com")
Περίπτωση 1:
![\[\( \alpha - 2 = 2\alpha - 7 \)\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-5bbf593f707cf3542ed337b8ab91e066_l3.png "Rendered by QuickLaTeX.com")
Σε αυτή την περίπτωση έχουμε:
![\[ \alpha - 2 = 2\alpha - 7. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-a494b1aaea54df6cbb654fccc3f0199d_l3.png "Rendered by QuickLaTeX.com")
Απλοποιούμε:
![\[ \alpha - 2\alpha = -7 + 2, \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-d358356fd528e500aca1c5ac6e139bbb_l3.png "Rendered by QuickLaTeX.com")
![\[ -\alpha = -5, \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-1257a4ba8597534e27343d2cac86923b_l3.png "Rendered by QuickLaTeX.com")
![\[ \alpha = 5. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-98be9062eb3b350a639fcb1c0a4cb037_l3.png "Rendered by QuickLaTeX.com")
Περίπτωση 2:
![\[\( \alpha - 2 = -(2\alpha - 7) \)\]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-20a609e35b7279aa2d8fe2de7770859e_l3.png "Rendered by QuickLaTeX.com")
Σε αυτή την περίπτωση έχουμε:
![\[ \alpha - 2 = -2\alpha + 7. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-f8839ef7db8c7cf5b17a256c8235f62e_l3.png "Rendered by QuickLaTeX.com")
Απλοποιούμε:
![\[ \alpha + 2\alpha = 7 + 2, \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-711b4103d88281341d21996337ace2e3_l3.png "Rendered by QuickLaTeX.com")
![\[ 3\alpha = 9, \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-b3fc5ec108de3bd3727250828c8cdf8b_l3.png "Rendered by QuickLaTeX.com")
![\[ \alpha = 3. \]](https://study4maths.gr/wp-content/ql-cache/quicklatex.com-cb2eeed35eb27f07733457c71100bff8_l3.png "Rendered by QuickLaTeX.com")
### Συμπέρασμα
Οι τιμές 
και 
είναι οι λύσεις που ικανοποιούν την αρχική εξίσωση.
ΑΣΚΗΣΕΙΣ ΓΙΑ ΕΞΑΣΚΗΣΗ
