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One of these expressions reduces to 1, and the other reduces to -1. Do you know which one is which? How do you know?

[tex]\[
\frac{x+3}{3+x} \quad \frac{3-x}{x-3}
\][/tex]


Sagot :

To determine which expression simplifies to 1 and which one simplifies to -1, let's examine each of the given expressions step-by-step:

### Expression 1
[tex]\[ \frac{x + 3}{3 + x} \][/tex]

Notice that the numerator [tex]\( x + 3 \)[/tex] and the denominator [tex]\( 3 + x \)[/tex] are actually the same algebraic expression because addition is commutative. Therefore:

[tex]\[ x + 3 = 3 + x \][/tex]

So the expression simplifies immediately:

[tex]\[ \frac{x + 3}{3 + x} = \frac{x + 3}{x + 3} = 1 \quad \text{(assuming } x + 3 \neq 0 \text{ which means } x \neq -3\text{)} \][/tex]

### Expression 2
[tex]\[ \frac{3 - x}{x - 3} \][/tex]

Here, observe that the numerator [tex]\( 3 - x \)[/tex] and the denominator [tex]\( x - 3 \)[/tex] are negatives of each other. Specifically:

[tex]\[ 3 - x = - (x - 3) \][/tex]

So we can rewrite the expression as:

[tex]\[ \frac{3 - x}{x - 3} = \frac{- (x - 3)}{x - 3} \][/tex]

When we simplify this, we get:

[tex]\[ \frac{- (x - 3)}{x - 3} = -1 \quad \text{(assuming } x - 3 \neq 0 \text{ which means } x \neq 3\text{)} \][/tex]

### Conclusion
- The expression [tex]\( \frac{x + 3}{3 + x} \)[/tex] simplifies to 1.
- The expression [tex]\( \frac{3 - x}{x - 3} \)[/tex] simplifies to -1.

Thus, we have determined which one is which:
- [tex]\( \frac{x + 3}{3 + x} = 1 \)[/tex]
- [tex]\( \frac{3 - x}{x - 3} = -1 \)[/tex]

And that's how we know!