Simplifying the Expressions

Oct 13, 2025 by miodeya.com 5 views

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

Understanding the Problem

When dealing with mathematical expressions, it's essential to understand the properties and behaviors of different operations. In this case, we're given two expressions:

${ \frac{x+3}{3+x} \quad \frac{3-x}{x-3} }$

Our goal is to determine which expression reduces to 1 and which one reduces to -1.

Q: What are the key concepts involved in this problem?

A: The key concepts involved in this problem are algebraic manipulation, properties of fractions, and understanding the behavior of expressions.

Q: How do we approach this problem?

A: To approach this problem, we need to simplify each expression by applying algebraic properties and manipulating the fractions.

Simplifying the Expressions

Let's start by simplifying the first expression:

${ \frac{x+3}{3+x} }$

We can simplify this expression by multiplying both the numerator and denominator by the conjugate of the denominator, which is (3-x).

\frac{x+3}{3+x} = \frac{(x+3)(3-x)}{(3+x)(3-x)}

Expanding the numerator and denominator, we get:

\frac{x+3}{3+x} = \frac{3x - x^2 + 9 - 3x}{9 - x^2}

Simplifying further, we get:

\frac{x+3}{3+x} = \frac{-x^2 + 9}{9 - x^2}

Now, let's simplify the second expression:

${ \frac{3-x}{x-3} }$

We can simplify this expression by multiplying both the numerator and denominator by the conjugate of the denominator, which is (x+3).

\frac{3-x}{x-3} = \frac{(3-x)(x+3)}{(x-3)(x+3)}

Expanding the numerator and denominator, we get:

\frac{3-x}{x-3} = \frac{3x + 9 - x^2 - 3x}{x^2 - 9}

Simplifying further, we get:

\frac{3-x}{x-3} = \frac{-x^2 + 9}{x^2 - 9}

Q: How do we determine which expression reduces to 1 and which one reduces to -1?

A: To determine which expression reduces to 1 and which one reduces to -1, we need to analyze the simplified expressions.

Looking at the first expression:

${ \frac{-x^2 + 9}{9 - x^2} }$

We can see that if x = 3, the expression reduces to:

${ \frac{-3^2 + 9}{9 - 3^2} = \frac{0}{0} }$

However, if x = -3, the expression reduces to:

${ \frac{-(-3)^2 + 9}{9 - (-3)^2} = \frac{0}{0} }$

But if we substitute x = 0, the expression reduces to:

${ \frac{-0^2 + 9}{9 - 0^2} = \frac{9}{9} = 1 }$

Therefore, the first expression reduces to 1 when x = 0.

Now, let's analyze the second expression:

${ \frac{-x^2 + 9}{x^2 - 9} }$

We can see that if x = 3, the expression reduces to:

${ \frac{-3^2 + 9}{3^2 - 9} = \frac{0}{6} = 0 }$

However, if x = -3, the expression reduces to:

${ \frac{-(-3)^2 + 9}{(-3)^2 - 9} = \frac{0}{-18} = 0 }$

But if we substitute x = 0, the expression reduces to:

${ \frac{-0^2 + 9}{0^2 - 9} = \frac{9}{-9} = -1 }$

Therefore, the second expression reduces to -1 when x = 0.

Conclusion

In conclusion, the first expression reduces to 1 when x = 0, and the second expression reduces to -1 when x = 0.

Q: What are the key takeaways from this problem?

A: The key takeaways from this problem are:

Algebraic manipulation is essential in simplifying expressions.
Understanding the properties of fractions is crucial in solving problems.
Analyzing the behavior of expressions is necessary to determine their values.

Q: How can we apply this knowledge to real-world problems?

A: We can apply this knowledge to real-world problems by:

Simplifying complex expressions to make them more manageable.
Understanding the behavior of expressions to make informed decisions.
Analyzing data to determine trends and patterns.

By applying this knowledge, we can solve a wide range of problems in mathematics, science, and engineering.