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4. \( 2
Ask by Daniels Marsh. in Turkey
Nov 23,2024

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Answer

\( 2a - b \) ifadesinin alabileceği en büyük tam sayı değeri 14'tür.

Solution

Verilen eşitsizlikler: \[ 2 < a < 7 \] \[ -1 < b < 3 \] İfademiz \( 2a - b \) ve bu ifadenin alabileceği en büyük tam sayıyı bulmamız gerekiyor. Öncelikle \( 2a \) ifadesini inceleyelim: \[ 2a \text{ için } 2 \cdot 2 < 2a < 2 \cdot 7 \implies 4 < 2a < 14 \] Yani \( 2a \) ifadesinin alabileceği değerler 4 ile 14 arasındadır. Şimdi \( -b \) ifadesini inceleyelim: \[ -b \text{ için } -(-1) > -b > -3 \implies 1 > -b > -3 \] Bu durumda \( -b \) ifadesinin alabileceği değerler -3 ile 1 arasındadır. Şimdi \( 2a - b \) ifadesini yazalım: \[ 2a - b = 2a + (-b) \] Burada \( 2a \) ifadesinin en büyük değeri 14 ve \( -b \) ifadesinin en büyük değeri 1'dir. Bu durumda: \[ 2a - b \text{ ifadesinin en büyük değeri } 14 + 1 = 15 \] Ancak, \( 2a - b \) ifadesinin tam sayı olması gerektiğini unutmayalım. Bu durumda, \( 2a - b \) ifadesinin alabileceği en büyük tam sayı değeri 15'ten küçük olmalıdır. Şimdi \( 2a - b \) ifadesinin alabileceği en büyük tam sayı değerini bulmak için \( 2a \) ve \( -b \) değerlerini en üst sınırda kullanmalıyız. Eğer \( a \) değerini 7'ye çok yakın bir değer alırsak (örneğin \( a = 6.999 \)), o zaman: \[ 2a \approx 2 \cdot 6.999 = 13.998 \] Eğer \( b \) değerini -1'e çok yakın bir değer alırsak (örneğin \( b = -0.999 \)), o zaman: \[ -b \approx -(-0.999) = 0.999 \] Bu durumda: \[ 2a - b \approx 13.998 + 0.999 = 14.997 \] Bu durumda \( 2a - b \) ifadesinin alabileceği en büyük tam sayı değeri 14 olacaktır. Sonuç olarak, \( 2a - b \) ifadesinin alabileceği en büyük tam sayı değeri: \[ \boxed{14} \]

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Verilen eşitsizliklere göre \(2a\) ifadesinin en büyük değeri için \(a\) değerinin en büyük sınırını alalım. Buradan \(a\) en fazla 7’ye çok yakın, fakat 7 olamaz. Dolayısıyla, \(2a\) ifadesinin en büyük değeri için \(a\) değerini 6.999... gibi bir değere yaklaştırmalıyız. Bu durumda: \[ 2a \approx 2 \cdot 6.999 \approx 13.998 \] Aynı şekilde, \(b\) ifadesinin en düşük sınırını alarak bu değeri çıkaralım. \(b\) en fazla 3’e çok yakın olmalıdır, fakat 3 olamaz. En düşük değer olarak -1’i alırsak: \[ -b \approx -(-1) = 1 \] Şimdi \(2a - b\) ifadesini birleştirirsek: \[ 2a - b \approx 13.998 + 1 = 14.998 \] En büyük tam sayı değeri bu durumda 14 olmaktadır. Yani, \(2a - b\) ifadesinin alabileceği en büyük tam sayı değeri 14'tür.

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