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Chapter 7: Problem 15

List all numbers for which each rational expression is undefined. $$ \frac{t-3}{t+8} $$

Short Answer

Expert verified
The rational expression is undefined for \( t = -8 \).

Step by step solution

01

- Identify the Rational Expression

The given rational expression is \( \frac{t-3}{t+8} \).
02

- Determine When the Denominator is Zero

The rational expression is undefined when the denominator is zero. Set the denominator equal to zero and solve for the variable \( t \). This gives: \( t + 8 = 0 \)
03

- Solve for the Variable

Solve the equation \( t + 8 = 0 \). Subtract 8 from both sides: \( t = -8 \)
04

- List the Values

The value of \( t \) that makes the rational expression undefined is \( t = -8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

undefined rational expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. To be considered valid, the denominator of a rational expression must not be zero, as dividing by zero is undefined in mathematics. When a rational expression has a denominator of zero, it cannot represent a real number.
denominator zero
For a rational expression to be meaningful, its denominator must not equal zero. If the denominator equals zero, the fraction becomes undefined. To find the values that make a denominator zero, set the denominator equal to zero and solve for the variable. In our example:
  • Expression: \(\frac{t-3}{t+8}\)
  • Denominator: \(t+8\)
  • Solve: \(t+8=0\)
  • Solution: \(t=-8\)
Therefore, the rational expression \(\frac{t-3}{t+8}\) is undefined when \(t=-8\).
solving equations
Solving equations, such as finding when the denominator of a rational expression is zero, involves finding the values that satisfy the equation. To solve an equation like \(t+8=0\), follow these steps:
  • Isolate the variable: Remove constants from the variable term by subtracting or adding them. For \(t+8=0\), we subtract 8 from both sides, resulting in \(t=-8\).
  • Check your solution: Verify if the solution makes the original equation true. Substitute \(t=-8\) back into the denominator \(t+8\) to see that it indeed equals zero: \(-8+8=0\), confirming that the expression is undefined for that value.
By solving equations like these, you can determine the values that make rational expressions undefined.

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