Problem 17 - 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 \(x \geq 5\). \(x-3>-2\)

Step-by-Step Solution

Answer

The interval is \((1, \infty)\).

Step 1: Understanding the Inequality

The given inequality is \(x - 3 > -2\). Our goal is to find the values of \(x\) that satisfy this inequality. This will help express the solution as an interval.

Step 2: Isolating the Variable

To solve for \(x\), we need to get it by itself on one side of the inequality. We can achieve this by adding 3 to both sides of the inequality. Doing so, we get: \[(x - 3) + 3 > -2 + 3\] which simplifies to \(x > 1\).

Step 3: Expressing as an Interval

The inequality \(x > 1\) implies that \(x\) can be any number greater than 1. Using interval notation, this set of numbers is expressed as \((1, \infty)\). This notation tells us that the interval starts just after 1 and goes on indefinitely to infinity.

Key Concepts

Interval Notation

Interval notation is a method used to describe a set of numbers along a number line. It provides a concise way to denote the range of values that satisfy an inequality. For instance, if you have a solution like \(x > 1\), the corresponding interval notation is \((1, \infty)\). This tells us that the set of solutions starts just after 1 and extends indefinitely to the right.

• Parentheses \(( )\) indicate that the endpoint is not included, usually used for strict inequalities like \(x > 1\).
• Brackets \([ ]\) imply that the endpoint is included, which corresponds to an inequality like \(x \geq 1\).

It is important to understand that the infinity symbol \(\infty\) always pairs with a parenthesis because infinity is a concept rather than a number, meaning it can never be "reached".

Solving Inequalities

Solving inequalities is similar to solving equations, with some key differences. An inequality compares two values or expressions, determining whether one is less than, greater than, less than or equal to, or greater than or equal to the other. Consider the inequality \(x - 3 > -2\).

• The first step is "Isolating the Variable" by manipulating the inequality so that \(x\) stands alone on one side. For this example, add 3 to both sides, leading to \(x > 1\).
• The inequality symbol remains the same unless you multiply or divide by a negative number, which reverses the inequality sign.
• Once isolated, express the solution using interval notation, a skill essential for developing a deeper understanding of ranges of values in algebra.

By mastering these steps, you can confidently approach inequalities, making them less intimidating and more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra, forming the core of problems involving inequalities.

• In the expression \(x - 3 > -2\), "\(x - 3\)" is an algebraic expression, where "\(x\)" is a variable representing unknown values.
• Your task in solving problems involving expressions is to manipulate these components to isolate the variable, helping you find the solution set.
• Understanding how to manipulate algebraic expressions is crucial, as it expands into more advanced topics in mathematics.

When you alter algebraic expressions, remember that each operation impacts the inequality. Gaining proficiency in this area enables you to tackle a wide array of mathematical challenges.