Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ (-5,8] $$

Step-by-Step Solution

Answer

The inequality is \(-5 < x \leq 8\).

Step 1: Understand Interval Notation

The given interval \((-5, 8]\) includes all numbers between -5 and 8, where -5 is not included (indicated by the parenthesis) and 8 is included (indicated by the square bracket). This means -5 is less than \(x\) and \(x\) is less than or equal to 8.

Step 2: Translate to Inequality

Using the information from the interval notation, we express the interval as the inequality. Since -5 is not included, we use \(x > -5\), and since 8 is included, we use \(x \leq 8\). So, the inequality is: \(-5 < x \leq 8\).

Key Concepts

Interval Notation

Interval notation is a way to describe sets of numbers on the number line, typically showing the range of values that a variable like \( x \) can take. In this notation, parentheses \((\)) and brackets \([]\) are used to indicate whether endpoints are included in the interval.

• Parentheses \((\) and \()\) mean that the endpoint is not included in the interval, depicting an "open" interval.
• Brackets \([\) and \()]\) mean that the endpoint is included, showing a "closed" interval.

Given the interval \((-5, 8]\), it can be read as: start at just above -5 and go up to and include 8.

This interval implies that \( x \) is any number greater than -5 but less than or equal to 8. Here, the parenthesis indicates that -5 is not part of the interval, whereas the bracket signifies 8 is included.

Inequalities

Inequalities represent the range of values that a variable can take but aren't pinned down to a fixed number like equations are. They define how numbers compare, using symbols such as "greater than," "less than," "less than or equal to," and "greater than or equal to."

• "<" means "less than"
• "\(\leq\)" means "less than or equal to"
• ">" means "greater than"
• "\(\geq\)" means "greater than or equal to"

In the context of \((-5, 8]\) as an inequality, we translate it to say:

• \(-5 < x\), meaning \( x \) is greater than -5 but not exactly -5.
• \( x \leq 8\), indicating that \( x \) can be up to and including 8.

Thus, the correct inequality form of our interval is \(-5 < x \leq 8\). This notation opens up the ability to solve for and analyze the behavior of \( x \) in algebraic contexts.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics and includes everything from solving equations to understanding the more complex structures involving simultaneous equations and inequalities.

In the provided solution, algebra assists in interpreting complex number sets like intervals through inequalities. This is a basic but vital first step to grasp how variables can interact and have limits.

• Translating between interval notations and inequalities is an essential algebraic skill, offering clear insights into how we understand constraints on a variable.
• Algebraic techniques allow us to effectively manage and solve inequalities, paving the way for more sophisticated problem-solving.

As you progress, mastering these interval translations will help simplify and handle more intricate algebraic expressions and their solutions efficiently.