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Inequalities are a fundamental part of mathematics that deal with the relative size of values. They are used to compare numbers and express relationships where quantities are not equal but instead greater or lesser than one another. In the given problem, we used an inequality to determine the values of \(x\) that make the function \(f(x)=\sqrt{x-5}\) valid.

An inequality involving a square root requires the expression inside the root to be zero or more because the square root of a negative number is not defined within the real number system.

• The symbol \(\geq\) stands for "greater than or equal to."
• Solve inequalities similarly to equations but remember that multiplying or dividing by a negative number reverses the inequality sign.

For this function, you start by setting \(x-5 \geq 0\), leading to \(x \geq 5\). This inequality tells us that the function value is only real when \(x\) is 5 or more.

Square roots are used to solve equations where a number is multiplied by itself to give a specific value. When solving equations involving square roots, it is essential to ensure that the expression inside the square root is non-negative in order to remain within the realm of real numbers.

The function \(f(x)=\sqrt{x-5}\) implies that the expression \(x-5\) must be zero or positive since square root values are not defined for negative numbers. To handle this correctly, use an inequality to set the inside of the square root greater than or equal to zero.

Here's why:

• The square root of negative numbers leads to complex numbers, which are not considered in this context.
• Ensuring the expression inside is positive preserves the integrity of real-valued outputs.

Understanding this lays the foundation for determining the domain of functions involving square roots.

Solving inequalities involves finding the set of all possible values of the variable that make the inequality true. It is a crucial step in defining the domain of functions, particularly those involving square roots and similar expressions.

For \(f(x)=\sqrt{x-5}\), you start by solving the inequality \(x-5 \geq 0\). To solve:

• Isolate \(x\) by adding 5 to both sides.
• This results in \(x \geq 5\).

Thus, the range of values for \(x\) that satisfies \(x-5 \geq 0\) is \(x \in [5, +\infty)\). In this interval notation, square brackets [ ] signify that the endpoint (in this case, 5) is included in the domain, while parentheses ( ) would indicate that it is not. Through solving this inequality, we define the domain of the function, ensuring it remains real and valid.