Problem 14 - 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 \(x \geq 5\). \((-5, \infty)\)

Step-by-Step Solution

Answer

The interval \((-5, \infty)\) can be expressed as \(x > -5\).

Step 1: Analyze the Given Interval

The given interval is \((-5, \infty)\). This is an open interval on the left side (indicated by the parenthesis '()'), and it extends to infinity on the right, which indicates that there is no upper bound.

Step 2: Determine the Inequality Symbol

Since the interval is open on the left at -5, the variable \(x\) is greater than -5. We use the inequality symbol \(>\) to express this relationship.

Step 3: Write the Inequality Expression

Given the analysis from the previous steps, we can express the interval \((-5, \infty)\) as the inequality \(x > -5\). This indicates that \(x\) can take any value greater than -5.

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers along a number line. It describes the endpoints of the interval and whether these endpoints are included in the set.

There are different types of intervals depending on whether the endpoints are included or excluded:

• Closed interval: Denoted with square brackets ([a, b]), both endpoints 'a' and 'b' are included in the interval, meaning the numbers are between and including 'a' and 'b'.
• Open interval: Denoted with parentheses ((a, b)), where neither endpoint is included. The numbers are between 'a' and 'b', but do not exactly touch 'a' or 'b'.
• Half-open (or half-closed) interval: It can be noted as [a, b) or (a, b], where one of the endpoints is included, and the other is not.
• Infinite interval: When the interval extends infinitely in one direction, it uses the infinity symbol (∞) with a parenthesis to indicate the lack of bounds in that direction, e.g., (a, ∞) or (-∞, b).

Interval notation succinctly captures the range of values within limits, providing a clear visual of number sets.

Bounds

In the context of inequalities and intervals, bounds define the limits between which the numbers in an interval lie. They can be either upper or lower, and they greatly determine how an interval is expressed.

• Lower bounds: The smallest number that is part of (or the beginning point of) an interval. For example, in the interval [2, 10], 2 acts as the lower bound.
• Upper bounds: The largest number that concludes (or the endpoint of) an interval. In the interval [2, 10], 10 is the upper bound.
• Infinite bounds: Sometimes intervals don't have a finite endpoint, such as (-∞, 5), where there is no lower bound.

Bounds help determine which numbers are solutions to an inequality, shaping the interval's form. They are critical in visualizing and solving algebraic inequalities.

Inequality Symbols

Inequality symbols are mathematical symbols used to compare values and express the relationship of sizes between different numbers. They play a fundamental role in interpreting intervals into manageable expressions.

• "Less than" (<) and "greater than" (>): Indicate that one value is smaller or larger than another. Open intervals utilize these as neither endpoint is included in the set.
• "Less than or equal to" (≤) and "greater than or equal to" (≥): These symbols show that a value is either smaller/larger or exactly equal to another. Closed intervals use these symbols at endpoints, showing inclusion.

Understanding these symbols is crucial as they provide precision in representing number sets in algebraic terms. They are the building blocks for forming equations and inequalities from intervals, bridging the concept of geometry with algebra.