Problem 70 - Algebra for College Students

Question

Find the indicated value for each given rational expression, if possible. $$T(x)=\frac{5-x}{x-5}, T(-9)$$

Step-by-Step Solution

Answer

The value of \( T(-9) \) is \( -1 \).

Step 1: Substitute the given value into the expression

The given rational expression is \( T(x) = \frac{5 - x}{x - 5} \). To find \( T(-9) \), substitute \( x = -9 \) into the expression:

Step 2: Simplify the numerator

Substitute \( x = -9 \) into the numerator: \( 5 - (-9) = 5 + 9 = 14 \).

Step 3: Simplify the denominator

Substitute \( x = -9 \) into the denominator: \( -9 - 5 = -14 \).

Step 4: Calculate the result

So the expression becomes \( T(-9) = \frac{14}{-14} \). Simplify this to get \( \frac{14}{-14} = -1 \).

Key Concepts

substitution in rational expressions

Substitution in rational expressions involves replacing the variable in an algebraic expression with a given value. In our example, we had the rational expression \( T(x) = \frac{5 - x}{x - 5} \). To find \( T(-9) \), we simply substituted \( x = -9 \) into the expression.
This process requires:

• Identifying the variable to substitute
• Replacing the variable with the given number

It's crucial to be careful with signs during substitution. In this case, negative signs play a key role. By correctly substituting and simplifying, we ensure the expression is evaluated accurately.

simplification of fractions

Simplification of fractions is essential for rational expressions. After substitution, we often get a fraction that needs to be simplified.
In our example, we had the fraction \( \frac{14}{-14} \). The numerator was calculated by substituting \( x = -9 \) into \( 5 - x \), resulting in \( 14 \). The denominator was determined by substituting \( x = -9 \) into \( x - 5 \), giving us \( -14 \).

The simplification process involves:

• Performing arithmetic operations like addition or subtraction
• Reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor

In this case, since 14 and -14 are the same number but with opposite signs, the simplified form is \( -1 \). Always check if the fraction can be reduced further to ensure accuracy.

negative values in algebra

Working with negative values in algebra requires careful attention, as they can significantly affect the outcome. When substituting negative values, it's important to properly handle the negative sign.
For instance, substituting \( x = -9 \) in the given expression \( T(x) = \frac{5 - x}{x - 5} \) involves:

• Carefully changing the signs during arithmetic operations
• Ensuring that subtraction of negatives results in correct positive values

Let's break it down:
1. The numerator calculation: \( 5 - (-9) \) converts to \( 5 + 9 = 14 \).
2. The denominator calculation: \( -9 - 5 = -14 \).
Since 14 divided by -14 gives us \( -1 \), we can see how handling negatives accurately ensures that the final result is correct. Remember, errors usually occur with signs, so double-check your work!