At IDNLearn.com, find answers to your most pressing questions from experts and enthusiasts alike. Join our Q&A platform to access reliable and detailed answers from experts in various fields.
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!
### 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!
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.