Question

The interval (2,5) can be written as the inequality _________________

Short answer

2 < x < 5

Step-by-step solution

Identify the Interval Type

The interval given is \( (2, 5) \), which represents all numbers between 2 and 5, not including 2 and 5 themselves. This is known as an open interval.

Write the Inequality Corresponding to the Interval

An open interval \((a, b)\) corresponds to the inequality \ a < x < b \, where \ x \ represents any number within the interval. For this problem, \(a = 2\) and \(b = 5\) . Therefore, the inequality is \ 2 < x < 5 \.

Key concepts

Open Interval

An open interval is a way of describing a range of values between two endpoints, where the endpoints themselves are not included in the range. This is visually represented using rounded parentheses like this: \( (a, b) \). For example, in the interval \( (2, 5) \), numbers like 2.1, 3, and 4.9 are included, but the numbers 2 and 5 are not.

This concept is important because it tells us which values are part of the solution set. If both endpoints were included, it would be a **closed interval** and represented with square brackets, like \( [2, 5] \). But for an open interval, only the numbers between the endpoints count.

Inequality Notation

Inequality notation is a mathematical way of expressing the range of values that are included in a set. This notation uses inequality symbols such as **<** (less than) and **>** (greater than) to show the boundaries of the range.

For the open interval \( (2, 5) \), the corresponding inequality notation is written as \[ 2 < x < 5 \]. This means that \ x \ can be any number greater than 2 and less than 5. The inequality tells us exactly where \ x \ can be positioned along the number line.

Understanding how to switch between interval notation and inequality notation is crucial. It helps you to better analyze and solve algebraic problems where you need to find a range of possible solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). These can be used to represent relationships and solve problems involving intervals.

When we express a range of values (like \( 2 < x < 5 \)) as an inequality, we’re actually working with an algebraic expression. The variables and the inequality signs together tell us about the range of solutions.

For example, if you solve an equation and find the solutions lie between two values, you’ll often express this in both interval and inequality notation. If you know how to translate between these notations, it will be easier to understand and solve more complex math problems.

Practice converting between different notations and solving related equations to strengthen your grasp of these algebraic concepts.