Question 52 of 58advanced🔍 AnalyzeShort Answer3 marks

Let aa and bb be two non-zero rational numbers such that a+1b=0a + \frac{1}{b} = 0. Without assigning any numerical values, determine whether abab is positive or negative. Justify your answer.

Correct Answer

Given that aa and bb are two non-zero rational numbers such that a+1b=0a + \frac{1}{b} = 0.

Step 1: Rearrange the given equation to isolate the variable aa on one side: a=−1ba = -\frac{1}{b}.

Step 2: Multiply both sides of the equation by bb. Since bb is a non-zero rational number, multiplication by bb is valid: a×b=−1b×ba \times b = -\frac{1}{b} \times b, which simplifies to ab=−1ab = -1.

Since the product abab equals −1-1, which is a negative rational number, abab is negative. Any non-zero rational number multiplied by its negative reciprocal will always yield a negative product.

Exercise: END-OF-CHAPTER EXERCISES | Q: 9 | (Page 25)
For More Understanding

Explanation

The problem requires determining the sign of the product abab based on the given equation a+1b=0a + \frac{1}{b} = 0. By rearranging the equation, we find that aa is the negative reciprocal of bb. Using the multiplication rule for rational numbers, multiplying aa by bb results in −1-1. Since −1-1 is negative, the product abab must be negative.

Solution Steps

  1. Step 1: Rearrange the given equation a+1b=0a + \frac{1}{b} = 0 to isolate aa, which gives a=−1ba = -\frac{1}{b}.

  2. Step 2: Multiply both sides by bb (since b≠0b \neq 0) to get ab=−1ab = -1.

  3. Step 3: Conclude that since ab=−1ab = -1, which is a negative number, abab is negative.

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