Problem 15 - Algebra for College Students

Question

Express each interval as an inequalit using the variable \(x\). For example, we can express the inter val \([5, \infty)\) as \(x \geq 5\). \([-7, \infty)\)

Step-by-Step Solution

Answer

The interval [-7, ∞) can be expressed as the inequality x ≥ -7.

Step 1: Understand the Interval Notation

The interval is given as \([-7, \infty)\). This means it starts at -7 and goes on to infinity. The square bracket [ indicates that -7 is included in the interval.

Step 2: Transform the Interval into an Inequality

Since the interval includes all numbers starting from -7 and extends to infinity, we express this as an inequality. All numbers \(x\) that are \(-7\) or greater belong to this interval.

Step 3: Write the Inequality

The inequality form for the interval \([-7, \infty)\) is \(x \geq -7\). This indicates that \(x\) can be -7 or any number greater than -7.

Key Concepts

Interval Notation

Intervals are a way of describing a continuous set of numbers. When we talk about intervals in mathematics, we refer to a range of values. In interval notation, the numbers that define the start and end of the interval are placed inside brackets. For example,

• An interval written as \([-7, \infty)\) indicates that the set includes all numbers from \-7\ to infinity.
• The use of a square bracket \([\) means that the boundary number is included in the interval.
• An interval would use a parenthesis \(]\) at infinity since infinity is not a specific number that can be included.

Hence, \([-7, \infty)\) means that our interval includes \-7\ and every number greater than \-7\. Interval notation is a concise and straightforward way to communicate an ongoing segment on the number line.

Inequality Transformation

The process of transforming an interval into an inequality is a step-by-step method that connects the graphical representation of intervals to algebraic expressions. This transformation helps us translate the inclusion and exclusion of numbers into a format that involves variables and comparison symbols.

• To transform \( [-7, \infty) \) into an inequality using the variable \(x\), we need to reflect the idea that all values starting at \-7\ can be represented as numbers that are equal to or greater than \-7\.
• The square bracket at \-7\ signifies that the interval "starts" precisely there, resulting in the inequality \(x \geq -7\).
• The infinity symbol is not included in the inequality since infinity is a concept rather than a number.

In inequality terms, this simply means we are looking at all possible values \(x\) such that \(x\) is greater than or equal to \-7\. This method ensures that the boundary conditions of the interval are clearly represented in a mathematical sentence.

Number Line Representation

Number line representation is a useful visual tool for understanding intervals and inequalities. It converts the algebraic concepts into easy-to-visualize graphics, which helps in grasping the range of possible values quickly.

• On a number line, numbers are positioned from left to right in increasing order.
• An interval like \([-7, \infty)\) would be shown starting at \-7\, with a filled dot or a bold line to show that \-7\ is included.
• The line would extend infinitely to the right, illustrating that all numbers greater than \-7\ are part of the interval.

This graphic representation makes it easy to see which numbers are included and how the inequality \(x \geq -7\) maps onto the number line. By visualizing the problem, one can often more readily understand the scope and limits of an interval or inequality.