Problem 58 - 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. $$ \sqrt{x+4} $$

Step-by-Step Solution

Answer

Question: Determine the restriction on the variable x so that the expression $$\sqrt{x+4}$$ represents a real number. Answer: The restriction on the variable x is $$x \ge -4$$.

Step 1: Identify the expression under the square root

The given expression is $$\sqrt{x+4}$$. The expression under the square root is $$x+4$$.

Step 2: Set the expression under the square root to be greater than or equal to zero

An expression under a square root must be greater than or equal to zero in order to represent a real number. So, we need to set the expression $$x+4$$ greater than or equal to zero: $$ x+4 \ge 0 $$

Step 3: Solve for x

Now, we need to solve the inequality $$x+4 \ge 0$$ for x: $$ x \ge -4 $$

Step 4: State the restrictions on x

In order for the expression $$\sqrt{x+4}$$ to represent a real number, the variable x must be greater than or equal to -4, which can be written as: $$ x \ge -4 $$ This is the restriction on the variable x.

Key Concepts

Understanding the Square Root

The square root is a mathematical function represented by the symbol \( \sqrt{} \). It essentially asks, "What number, when multiplied by itself, gives the original number inside the root?" For example, the square root of 9 is 3, because multiplying 3 by itself, i.e., 3 x 3, gives 9. This concept is straightforward when dealing with positive numbers. However, when the expression inside the square root becomes negative, it leads to complex numbers, which go beyond the scope of real numbers and into the realm of imaginary numbers.

In this exercise, the focus is on ensuring that the number inside the root, \( x+4 \), remains non-negative (either zero or positive). This is because a square root can only yield a real number if its argument is not negative. So, when dealing with expressions like \( \sqrt{x+4} \), we need to identify conditions where \( x+4 \ge 0 \) to maintain real number solutions.

Exploring Inequalities

Inequality is a comparison between two expressions that may not be equal. In mathematics, inequalities are represented using symbols like \( > \), \( < \), \( \ge \), and \( \le \), which stand for "greater than", "less than", "greater than or equal to", and "less than or equal to," respectively.

In the provided exercise, solving the inequality \( x+4 \ge 0 \) helps determine the values of \( x \) that make the expression inside the square root valid for real numbers. To solve this, subtract 4 from both sides to isolate \( x \). This results in \( x \ge -4 \).

Understanding inequalities and how to solve them is crucial as they frequently appear in various mathematical problems. They help define ranges and domain restrictions for functions and expressions to maintain their validity in real number mathematics.

Decoding Mathematical Expressions

Mathematical expressions are combinations of numbers, operators, variables, and sometimes functions. They can range from simple equations to complicated algebraic statements. Each part of an expression has a specific role in defining its overall structure and result.

Take \( \sqrt{x+4} \) as an example. Here we have a square root function applied to an algebraic expression \( x+4 \). Each part interacts mathematically to determine if the expression holds particular qualities, such as being part of the set of real numbers.

• The variable \( x \) can take on different values.
• The operation inside the square root (i.e., \( +4 \)) effectively shifts the "starting point" of possible values.

Understanding these segments not only clarifies what each component contributes but also aids in solving or simplifying the expression effectively. Breaking down and translating expressions is an essential skill across all levels of math, from primary arithmetic to advanced calculus.