Verified Answer :

  • The term \(\sqrt{x-3}\) requires \(x \geq 3\).
  • The term \(\sqrt{x+1}\) requires \(x \geq -1\).
  • The term \(\frac{3}{5-x}\) requires \(x eq 5\).
  • The intersection of these conditions defines the domain as \(\boxed{[3, 5) \cup (5, \infty)}\).

Explanation

  1. Analyzing the Components

To determine the domain of the equation \(\sqrt{x-3} + \sqrt{x+1} = \frac{3}{5-x}\), we must consider each component that restricts the possible values of \(x\). The domain consists of all real numbers \(x\) that allow every part of the expression to be well-defined.

  1. Evaluating Individual Constraints
  2. The square root \(\sqrt{x-3}\) is defined for all real numbers where the radicand is non-negative:

\(x - 3 \geq 0 \implies x \geq 3\).

  1. The square root \(\sqrt{x+1}\) is defined for all real numbers where the radicand is non-negative:

\(x + 1 \geq 0 \implies x \geq -1\).

  1. The rational expression \(\frac{3}{5-x}\) is defined as long as the denominator is not zero:

\(5 - x eq 0 \implies x eq 5\).

  1. Combining Constraints and Finding the Domain

To find the domain of the entire equation, we need the intersection of all conditions identified:

  • From condition (1), we require \(x \geq 3\).
  • From condition (2), we require \(x \geq -1\).
  • Combining these, the intersection of \(x \geq 3\) and \(x \geq -1\) is simply \(x \geq 3\).
  • Finally, we must satisfy condition (3), which is \(x eq 5\).

Combining \(x \geq 3\) and \(x eq 5\), the valid range for \(x\) is all numbers greater than or equal to 3, excluding 5. In interval notation, this is \([3, 5) \cup (5, \infty)\).

Examples

Finding the domain of an equation is like determining the operating conditions for a machine. Imagine a machine that can only process material if it has at least a certain temperature and pressure, but will break if the pressure exceeds a specific threshold. By identifying these limitations, you ensure the machine operates safely and effectively. Similarly, in mathematics, identifying the domain ensures that we are only working with values that allow the equation to exist within the set of real numbers.

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