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Zorluk: ZorFractions and Rational Numbers

Let aa and bb be rational numbers such that −1<a<0<b<1-1 < a < 0 < b < 1. Which of the following expressions MUST be strictly greater than 11? Select all such expressions.

  1. b−ab\frac{b - a}{b}Cevap
  2. 1+b1+a\frac{1 + b}{1 + a}Cevap
  3. C
    a+bb\frac{a + b}{b}
  4. D
    b−a1−a\frac{b - a}{1 - a}
  5. E
    1b−a\frac{1}{b - a}

Cevap

The expressions b−ab\frac{b - a}{b} and 1+b1+a\frac{1 + b}{1 + a} MUST be strictly greater than 1.
The expression representing the difference (b−a)(b - a) divided by bb simplifies to 1+−ab1 + \frac{-a}{b}. Because −a-a and bb are both positive, this term is strictly greater than 1. Similarly, the expression 1+b1+a\frac{1 + b}{1 + a} divides a numerator greater than 1 by a positive denominator less than 1, which always yields a value strictly greater than 1.

Adım Adım Çözüm

1
Analyze the expression b−ab\frac{b - a}{b} by splitting the fraction.
b−ab=bb−ab=1−ab\frac{b - a}{b} = \frac{b}{b} - \frac{a}{b} = 1 - \frac{a}{b}.
Since a<0a < 0 and b>0b > 0, the quotient ab\frac{a}{b} is strictly negative. Subtracting a negative number from 1 is equivalent to adding a positive number, so 1−ab>11 - \frac{a}{b} > 1.
2
Analyze the expression 1+b1+a\frac{1 + b}{1 + a} by establishing bounds for the numerator and denominator.
Since b>0b > 0, 1+b>11 + b > 1. Since −1<a<0-1 < a < 0, adding 1 yields 0<1+a<10 < 1 + a < 1.
Dividing any real number greater than 1 by a positive real number less than 1 produces a value strictly greater than 1.
3
Evaluate the remaining options with counterexamples or algebraic bounds to verify they are not guaranteed to be greater than 1.
The expression a+bb=1+ab<1\frac{a + b}{b} = 1 + \frac{a}{b} < 1; the expression b−a1−a<1\frac{b - a}{1 - a} < 1 because b<1  ⟹  b−a<1−ab < 1 \implies b - a < 1 - a; and 1b−a\frac{1}{b - a} can be less than 1 when b−a>1b - a > 1 (e.g., b=0.8,a=−0.5  ⟹  b−a=1.3b = 0.8, a = -0.5 \implies b - a = 1.3).
Demonstrating that an expression is less than 1 or can be less than 1 eliminates it from being strictly greater than 1 for all valid rational numbers aa and bb.

Anahtar Kavram

Properties and Inequalities of Rational Numbers
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Konular

Fractions and Rational Numbers

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