Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ [0,4) $$

Step-by-Step Solution

Answer

The inequality is \(0 \leq x < 4\).

Step 1: Analyze the Interval Notation

The interval notation \([0,4)\) indicates that the interval includes 0 but does not include 4. The square bracket \([ \) denotes inclusion, while the parenthesis \( ) \) denotes exclusion.

Step 2: Translate Interval to Inequality

To express the interval \([0,4)\) as an inequality in the variable \(x\), we need to capture the conditions defined by the interval. Since 0 is included, we write \(x \geq 0\). Since 4 is not included, we write \(x < 4\). Combining both parts, we get the inequality: \(0 \leq x < 4\).

Key Concepts

Interval Notation

Interval notation is a convenient way to express a range of numbers, often used in mathematics to describe a set of values, all of which fall between two endpoints. For example, the interval \([0,4)\) represents all numbers starting from 0 and going up to, but not including, 4. It's important to note the different types of brackets used:

• A square bracket \([\) indicates that the endpoint is included in the interval. This is known as a 'closed interval'.
• A parenthesis \(()\) signals that the endpoint is not included, known as an 'open interval'.

Hence, when you see \([0,4)\), it means 0 is included but 4 is not. This is especially important in calculus, algebra, and other math disciplines where these concepts are heavily utilized.

Inequality Symbols

Inequality symbols are used to compare values, showing whether one number is less than, greater than, equal to, or not equal to another number. In mathematics, these symbols are crucial for solving equations involving ranges or constraints. Common inequality symbols include:

• \(<\) - Less than
• \(>\) - Greater than
• \(\leq\) - Less than or equal to
• \(\geq\) - Greater than or equal to

In the context of our interval \([0,4)\), the inequality symbols \(x \geq 0\) and \(x < 4\) express the range of values for \(x\). This means the variable \(x\) can take any value from 0 to just before 4. These symbols help precisely define boundaries for solutions to equations or inequalities.

Variable Expression

A variable expression is a mathematical phrase that involves variables, numbers, and operation symbols. It represents an unknown value and is a fundamental component of algebra. Variables are often denoted by letters like \(x\), \(y\), or \(z\), and they allow us to write general rules for mathematics that apply to many different numbers.
In the inequality \(0 \leq x < 4\), \(x\) is the variable, representing any real number that satisfies the inequality. This is a perfect example of how variable expressions are used to convey solutions to mathematical problems involving intervals.
When you manipulate variable expressions, you can solve for specific values by applying arithmetical operations and adhering to the rules of algebra. They are invaluable in expressing relationships between numbers and in solving real-world problems by creating models to predict various scenarios.