One Of These Expressions Reduces To 1 And The Other Reduces To -1. Do You Know Which One Is Which? How Do You Know?$\[ \frac{x+3}{3+x} \quad \frac{3-x}{x-3} \\]Determine The Value Each Expression Reduces To:$\[ \frac{x+3}{3+x} = \square

by ADMIN 237 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:

x+33+x3−xx−3{ \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:

x+33+x{ \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:

3−xx−3{ \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:

−x2+99−x2{ \frac{-x^2 + 9}{9 - x^2} }

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

−32+99−32=00{ \frac{-3^2 + 9}{9 - 3^2} = \frac{0}{0} }

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

−(−3)2+99−(−3)2=00{ \frac{-(-3)^2 + 9}{9 - (-3)^2} = \frac{0}{0} }

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

−02+99−02=99=1{ \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:

−x2+9x2−9{ \frac{-x^2 + 9}{x^2 - 9} }

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

−32+932−9=06=0{ \frac{-3^2 + 9}{3^2 - 9} = \frac{0}{6} = 0 }

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

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

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

−02+902−9=9−9=−1{ \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.