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!
