Problem 57 - 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. $$ \sqrt{y+10} $$

Step-by-Step Solution

Answer

Answer: The proper restriction on the variable y is $$y \ge -10$$.

Step 1: Identify the square root

We are given the expression: $$ \sqrt{y+10} $$ In this expression, we have the square root of (y+10).

Step 2: Restrict the value inside the square root

For the expression to be a real number, the value inside the square root must be greater than or equal to 0. So we need to set the value inside the square root (y+10) greater than or equal to 0: $$ y+10 \ge 0 $$

Step 3: Solve the inequality

Now, we need to solve this inequality to get the proper restriction for y: $$ y \ge -10 $$ Therefore, the proper restriction on the variable y so that the expression represents a real number is: $$ y \ge -10 $$

Key Concepts

Understanding the Square Root

A square root refers to a special value that, when multiplied by itself, gives the original number. In the context of algebra, the square root notation \(\sqrt{x}\) often appears. It's crucial to understand that the square root of a number is only defined for non-negative numbers when we are working within the real number system. This means the expression \( \sqrt{y+10} \) requires the term \( y+10 \) to be a non-negative number (greater than or equal to zero) for the square root to produce a real number. In practical terms, when you see a square root in an algebraic expression, it's a signal to ensure whatever is inside it is zero or positive. This ensures the mathematics used remains in the realm of real numbers, avoiding imaginary or complex numbers, which are not covered in most basic algebra courses.

Exploring Real Numbers

Real numbers include both positive and negative numbers, zero, and all the decimal and fractional numbers. Put simply, if a number can be located on the continuous number line, it is a real number. The significance of real numbers in algebraic expressions is immense as they form the basis for most of the calculations in arithmetic and algebra. When dealing with expressions involving a square root, as in \( \sqrt{y+10} \), we must ensure that all numbers inside the square root are real numbers to avoid shifting to imaginary or complex numbers. Imaginary numbers involve the square root of negative numbers, something which isn't permitted under the real number system. Hence, to keep an expression real, values like \( y+10 \) must remain non-negative.

Demystifying Inequalities

Inequalities are expressions that denote a relationship between two values when they are not equal. They are often represented using symbols like \(>\), \(<\), \(\ge\), and \(\le\). Understanding inequalities is essential because they tell us which values a variable can or cannot take. When we see \( y + 10 \ge 0 \), it indicates the values of \( y \) must be such that \( y+10\) is always zero or positive.

• The inequality \( \ge \) means "greater than or equal to," so \( y \) cannot be less than \(-10\).
• Solving the inequality involves isolating \( y \), leading to \( y \ge -10 \), indicating the smallest possible value \( y \) can take.

Thus, understanding inequalities helps in determining the acceptable values of variables to fulfill the given conditions of an equation or expression. This understanding is vital in ensuring the solution remains within the limits of the real number system.