What is wrong with the following proof. You must explain your answer in words.
a>3
3a>3(3)
3a-a^2>9-a^2
a(3-a)>(3-a)(3+a)
a>3+a
0>3
the problem with the proof is that the fourth step needs the sign to be reverse...but why?
i looked at step 3 and there is nothing negative to reverse the sigh for...
From the first statement, a must be a positive number. In fact, it must exceed 3. (This is very important later). The second and third statements are correct, since you are either multiplying by positive numbers or just factoring. The fourth statement is incorrect because 3-a is a negative number.The directon of the > sign must change.
The problem with the proof lies in step 3, particularly with the inequality being reversed. To understand why the sign needs to be reversed, let's take a closer look at steps 2 and 3.
In step 2, we have 3a > 3(3), which simplifies to 3a > 9. This is a valid inequality since we are multiplying both sides by a positive number (3 in this case).
However, in step 3, the inequality 3a - a^2 > 9 - a^2 is manipulated. Notice that we subtracted a^2 from both sides. The issue arises because we are assuming that a^2 is positive. But what if a is a negative number?
Let's consider two cases:
Case 1: a > 0
In this case, a^2 is positive, and subtracting a^2 from both sides of the equation does not affect the direction of the inequality. The inequality remains the same.
Case 2: a < 0
In this case, a^2 is still positive since squaring a negative number gives a positive result. However, when we subtract a^2 from both sides, we need to reverse the direction of the inequality. This is because we are subtracting a positive number from both sides, which shifts the relative positions of the values.
Therefore, when manipulating inequalities with squared terms, it is essential to consider both positive and negative values for the variable. In step 3, the proof assumed that a is positive, leading to an incorrect reversal of the inequality sign. The correct step should instead be:
3a - a^2 < 9 - a^2
By using the correct inequality, the proof will yield a valid result.