Problem 60 - Elementary Algebra

Question

For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number. $$ \sqrt{h-11} $$

Step-by-Step Solution

Answer

Question: Determine the restrictions on the variable h for the expression $$\sqrt{h - 11}$$ to represent a real number. Answer: The variable h must be greater than or equal to 11.

Step 1: Identify the expression inside the square root

In this case, we have an expression inside the square root, which is $$h - 11$$.

Step 2: Determine what values will make the expression non-negative

To find which values of h will make the expression $$h - 11$$ non-negative, set the expression greater than or equal to zero: $$(h - 11) \geq 0$$.

Step 3: Solve the inequality

Now we will solve the inequality to find the values of h that satisfy the condition: Add 11 to both sides of the inequality: $$h \geq 11$$. So, the proper restriction on the variable h is that it must be greater than or equal to 11 to ensure that the expression represents a real number.

Key Concepts

Real Numbers

Real numbers encompass a wide range of numbers, including integers, fractions, and irrational numbers. Essentially, they represent any point along a continuous number line. Although they seem simple, real numbers are crucial in various mathematical contexts.

When dealing with expressions like \(\sqrt{h-11}\), we aim to ensure the result is a real number. This requirement means the number under the square root must not lead to negative outcomes, as square roots of negative numbers are not considered real numbers.

• Integers: Whole numbers, both positive and negative, including zero.
• Fractions: Numbers represented in the form of \(\frac{a}{b}\), where \(b eq 0\).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or \(\sqrt{2}\).

Thus, ensuring a real number result is vital for accurate computations and applications.

Inequalities

Inequalities show how one expression relates to another and are integral to solving problems in algebra. In our exercise, the inequality \( (h - 11) \geq 0 \) is key to finding the values of \( h \) that ensure the expression under the square root leads to a real number.

Here's a simple breakdown of working with inequalities:

• "Greater than or equal to" (\(\geq\)): Includes both the greater value and the number itself.
• "Less than or equal to" (\(\leq\)): Includes both the lesser value and the number itself.
• "Greater than" (\(>\)) or "Less than" (\(<\)): Only considers values strictly larger or smaller.

Using inequalities, as in the exercise, allows us to pinpoint valid values of \( h \), facilitating computations while maintaining mathematical integrity.

Square Roots

Square roots are both fundamental and fascinating in algebra. They involve finding a number, which when multiplied by itself, equals the original number. For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

In expressions like \(\sqrt{h-11}\), ensuring that the expression inside the square root is non-negative allows the result to be a real number.

• Non-Negative Requirement: The expression within the square root must be zero or positive.
• Imaginary Numbers: If the number inside is negative, the result falls in the realm of complex numbers, not applicable in ensuring real number outcomes.
• Calculation: Solve any internal expression to determine the values where a real square root is possible.

Adhering to these principles ensures that solutions are mathematically valid and applicable in real-world contexts.