Problem 32 - Algebra for College Students

Question

Solve each inequality. $$(x+9)^{2} \geq 0$$

Step-by-Step Solution

Answer

The inequality holds for all real numbers: \\( x \\in \\mathbb{R} \\).

Step 1: Understand the Inequality

The inequality \( (x+9)^2 \geq 0 \) states that the square of \( (x+9) \) is greater than or equal to zero. A square of any real number is always non-negative.

Step 2: Analyze the Square

For any real number \(a\), \(a^2\) is always greater than or equal to zero. Therefore, \( (x+9)^2 \geq 0 \) is always true regardless of the value of \( x \).

Step 3: Determine the Solution Set

Since a square is always non-negative for any real number, the inequality holds for any \( x \). Hence, the solution set includes all real numbers, denoted by \( x \in \mathbb{R} \).

Key Concepts

Real Numbers

Real numbers are the building blocks of everyday calculations in mathematics. They encompass a vast collection of numbers that can be both positive and negative, whole or fractions. Simply put, real numbers include all the numbers you can think of on the number line.

• Integers like -3, 0, and 5 are real numbers.
• Fractions such as 1/2 and -7/8 are also real numbers.
• Even those tricky irrational numbers, like \( \sqrt{2} \) or \( \pi \), fall into this category.

So, when we say the solution set of an inequality includes all real numbers, we essentially mean any number you can possibly think of—excluding those imaginary numbers like \( \sqrt{-1} \). This vast set is signified by the symbol \( \mathbb{R} \) (a fancy mathematical way to say "real numbers").

Square of a Number

When discussing the square of a number, it's important to understand that squaring involves multiplying the number by itself. Mathematically, if you have a number \( a \), then the square of \( a \) is represented as \( a^2 \).

• For example, \( 3^2 = 9 \), because \( 3 \times 3 = 9 \).
• Squares of negative numbers are positive, e.g., \( (-3)^2 = 9 \).
• The square of zero is zero itself, \( 0^2 = 0 \).

An interesting aspect of squaring numbers is that the result is always non-negative. That's because multiplying two negative numbers or two positive numbers always results in a positive number, while zero remains zero. This property is crucial in solving inequalities like \( (x+9)^2 \geq 0 \). That means no matter what value \( x \) takes, \( (x+9)^2 \) is never negative.

Inequality Solution Set

An inequality solution set is the range of values that satisfy the conditions of an inequality. In the inequality \( (x+9)^2 \geq 0 \), we're interested in finding out for which values of \( x \) this inequality is true.

To figure this out, we need to consider the nature of the square of a number (as discussed earlier). Since the square of any expression, such as \( (x+9)^2 \), is non-negative for any \( x \), it implies:

• The inequality \( (x+9)^2 \geq 0 \) holds true for all \( x \).
• Thus, the solution set includes every possible real number.
• Therefore, the solution can be expressed as \( x \in \mathbb{R} \).

This solution set tells us that no matter which real number we choose for \( x \), the inequality will always be satisfied. It's a concept that underscores the all-encompassing and reassuring reliability of real numbers when squared in inequalities.