Problem 56 - Algebra for College Students

Question

Solve each inequality. State the solution set using interval notation when possible. \(x^{2} \geq 0\)

Step-by-Step Solution

Answer

The solution set is \((-\infty, \infty)\).

Step 1 - Understand the Inequality

The given inequality is \(x^{2} \geq 0\). This means we have to find all the values of \(x\) for which the square of \(x\) is greater than or equal to zero.

Step 2 - Analyze the Expression

For any real number \(x\), \(x^{2}\) (the square of \(x\)) is always non-negative. This is because squaring any real number, whether positive, negative, or zero, results in a value that is zero or positive.

Step 3 - Identify the Solution Set

Since \(x^{2}\) is always non-negative for all real numbers, the inequality \(x^{2} \geq 0\) holds true for all real numbers \(x\).

Step 4 - Write the Solution in Interval Notation

The set of all real numbers is represented in interval notation as \((-\infty, \infty)\). This indicates that the inequality holds for all values from negative infinity to positive infinity.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions involving the symbols \(<, \leq, >, \geq\). They are used to show that one value is larger or smaller than another. When solving inequalities, our goal is to find all possible values of the variable that make the inequality true. For example, consider the inequality \(x^{2} \geq 0\). Here, we need to find out for which values of \(x\) the square of \(x\) is greater than or equal to zero.

Types of Inequalities:

• Strict Inequalities: These include \(<\) and \(>\), indicating that one quantity is strictly less than or greater than another.
• Non-Strict Inequalities: These include \(\leq\) and \(\geq\), which also allow for equality.

Solving Inequalities:

• Isolate the variable: Try to get the variable on one side of the inequality.
• Consider the direction of the inequality: When multiplying or dividing by a negative number, the direction of the inequality changes. For example, if you multiply both sides of the inequality \(a < b\) by -1, it becomes \(-a > -b\).

Real Numbers

Real numbers are a broad category of numbers that include all the numbers on the number line. They encompass several subsets:

• Natural Numbers: The counting numbers: 1, 2, 3, ...
• Whole Numbers: Natural numbers plus zero: 0, 1, 2, 3...
• Integers: Whole numbers and their negatives: -2, -1, 0, 1, 2...
• Rational Numbers: Numbers that can be expressed as a fraction or ratio of two integers: 1/2, 2/3, etc.
• Irrational Numbers: Numbers that cannot be written as a simple fraction. Their decimal representation is non-repeating and non-terminating: \(\pi, \sqrt{2}\), etc.

In the context of inequalities, when we say the solution holds for all real numbers, we mean that there are no exceptions within the real numbers. For instance, in the inequality \(x^{2} \geq 0\), since squaring any real number always results in a non-negative number, the inequality is true for all real numbers.

Interval Notation

Interval notation is a way to describe sets of numbers along a number line. It uses parentheses and brackets to show where the intervals start and end.

Types of Notations:

• Parentheses \(( )\) : Used when the endpoint is not included in the interval. For example, \((1, 3)\) means greater than 1 and less than 3, but not including 1 and 3.
• Brackets \([ ]\) : Used when the endpoint is included in the interval. For example, \([1, 3]\) means all numbers from 1 to 3, including 1 and 3.

Mixed Notation:

• Sometimes, you need both parentheses and brackets. For example, \((1, 3]\) means greater than 1 but less than or equal to 3.

In the given problem \(x^{2} \geq 0\), no matter what real number you choose for \(x\), squaring it will always give you zero or a positive number. Thus, the solution set includes all real numbers, which is expressed in interval notation as \((-\textrm{∞}, \textrm{∞})\).