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QuestionIdentify which of the following expressions equals 1 and which equals -1:
x+33+x3−xx−3 \frac{x+3}{3+x} \quad \frac{3-x}{x-3}

Studdy Solution

STEP 1

Assumptions1. We are given two expressions x+33+x\frac{x+3}{3+x} and 3−xx−3\frac{3-x}{x-3}. . We are asked to determine which of these expressions reduces to1 and which reduces to -1.

STEP 2

Let's start by simplifying the first expression x++x\frac{x+}{+x}.
This expression is already in its simplest form. The numerator and the denominator are the same, just written in a different order. Therefore, this expression simplifies to1 for all x except - (as division by zero is undefined).
x++x=1\frac{x+}{+x} =1

STEP 3

Now, let's simplify the second expression 3−xx−3\frac{3-x}{x-3}.
At first glance, it might seem that this expression is also equal to1, since the numerator and the denominator are the same, just written in a different order. However, the order of subtraction matters in mathematics.

STEP 4

To make the numerator and the denominator of the second expression look the same, we can factor out a -1 from the numerator.3−xx−3=−(x−3)x−3\frac{3-x}{x-3} = \frac{-(x-3)}{x-3}

STEP 5

Now, the numerator and the denominator are the same, just with a negative sign in front of the numerator. Therefore, this expression simplifies to -1 for all x except3 (as division by zero is undefined).
−(x−3)x−3=−1\frac{-(x-3)}{x-3} = -1So, the expression x+33+x\frac{x+3}{3+x} reduces to1 and the expression 3−xx−3\frac{3-x}{x-3} reduces to -1.

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