Problem 24 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is all real numbers: \(x \in \mathbb{R}\).

Step 1: Identify the expression

The inequality to solve is \((x+9)^2 \geq 0\). This is a square of a binomial expression.

Step 2: Understand properties of squares

For any real number \(a\), the square \(a^2\) is always greater than or equal to zero, since squaring any real number results in a non-negative number.

Step 3: Apply the property

Since \((x+9)^2 \geq 0\) is a perfect square, the inequality holds for all real numbers \(x\). Perfect squares are zero or positive for any real number.

Step 4: Identify if equality holds

The equality \((x+9)^2 = 0\) holds when \(x+9 = 0\), or \(x = -9\). At this point, the inequality \((x+9)^2 \geq 0\) becomes an equality.

Step 5: Determine the solution set

Since the inequality is satisfied for all real numbers by step 2, and equality holds at \(x = -9\), the solution set is all real numbers \(x \in \mathbb{R}\).

Key Concepts

Real Numbers

Real numbers include all the numbers that can be found on the number line. This encompasses every number you can think of, including:

• Whole numbers
• Integers
• Fractions
• Decimals
• Irrational numbers (like \( \sqrt{2} \))

Real numbers can be negative, positive, or zero. An important aspect of real numbers in inequalities is that when you square any real number, the result is always zero or positive. That is because multiplying two positive numbers or two negative numbers always yields a positive result, and zero squared is zero. This property leads us to understand that when dealing with inequalities like \((x+9)^2 \geq 0\), the zero or positive result holds true for any real number \(x\).

Binomial Expression

A binomial expression is a type of algebraic expression that contains exactly two terms separated by a plus (\(+\)) or minus (\(-\)) sign. In the inequality \((x+9)^2 \geq 0\), the expression \(x+9\) is a binomial. A binomial expression can look like these:

• \(x+9\)
• \(a-b\)
• \(3y + 4z\)

To solve inequalities involving binomials, one often needs to perform operations such as expanding or factoring the binomial. However, in this particular inequality, since the binomial is squared and greater than or equal to zero, it directly utilizes the property of squares, which simplifies solving the inequality. Thus, understanding binomials paves the way to dealing with expressions in more complex algebraic equations and inequalities.

Perfect Square

A perfect square in algebra occurs when you multiply a binomial by itself. For any number or expression, squaring it means writing it in the form of \(a^2\), and the result is always non-negative. In the inequality \((x+9)^2 \geq 0\), \((x+9)^2\) is a perfect square. The key quality of perfect squares is their non-negative value for all real numbers. For any real number \(x\), if \((x+9)^2\) is computed, the smallest possible value is \(0\), which occurs when \(x+9 = 0\), or equivalently, \(x = -9\). Typically, perfect squares are useful for confirming the solution set of such inequalities is inclusive of all real numbers, which means the expression satisfies the inequality across the entire number line.

Solution Set

The solution set in an inequality refers to all the possible values of the variable that satisfy the inequality condition. For the inequality \((x+9)^2 \geq 0\), the property of perfect squares tells us that this condition will be met for any real number, hence no number is excluded.To identify the solution set:- Recognize that because squaring any expression or number cannot be negative, \((x+9)^2\) will never be less than zero.- The expression equals zero at \(x = -9\).- Therefore, since the inequality holds for all real numbers, the solution set is \(x \in \mathbb{R}\).This means every real number is a solution to the inequality, including both positive and negative numbers, as well as fractions and irrational numbers. Understanding solution sets is crucial because it tells you the range of values that can be plugged into the original inequality without violating its conditions.