Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Express the inequality \(3 \leq x<8\) using interval notation.

Short answer

[3, 8)

Step-by-step solution

- Understand the inequality

The inequality provided is composed of two parts: the lower bound and the upper bound. The lower bound is represented by the inequality 3 ≤ x, meaning that x is greater than or equal to 3. The upper bound is represented by the inequality x < 8, meaning that x is less than 8.

- Identify the interval notation symbols

Interval notation uses brackets and parentheses to denote inclusion or exclusion of endpoints. A square bracket [ or ] indicates that the endpoint is included, while a parenthesis ( or ) indicates that the endpoint is not included.

- Convert the lower bound

The lower bound 3 ≤ x means that the value 3 is included. Therefore, we use a square bracket for the lower bound: [3

- Convert the upper bound

The upper bound x < 8 means that the value 8 is not included. Therefore, we use a parenthesis for the upper bound: 8)

- Combine the bounds into interval notation

Combine the lower and upper bounds using a comma to separate them. Therefore, the interval notation for the inequality 3 ≤ x < 8 is [3, 8).

Key concepts

Inequalities

Inequalities are a fundamental concept in algebra that describe the relationship between two values. Unlike equations, which show equality, inequalities show whether one value is greater than, less than, or not equal to another value. They are typically expressed using symbols like <, >, ≤, and ≥. This allows us to describe a range of possible values for a variable, rather than a single specific value.
Inequalities can be open or closed, depending on whether the endpoints are included in the set of possible values.
For example, in the inequality \(3 \leq x<8\), 3 is included while 8 is not. This can be very useful when you need to express conditions or constraints in both math problems and real-world scenarios, such as determining the acceptable dosage of medicine or specifying time intervals for events.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is the language through which we describe mathematical relationships. In the context of inequalities, algebra helps us understand how different numbers relate to each other.
For example:
- When solving an inequality like \(3 \leq x<8\), you identify the range of numbers x can take.
- Through algebra, we learn to combine, rearrange, and solve these inequalities through various methods.
Many algebraic techniques are used to manipulate and solve inequalities, including:

• Adding or subtracting the same value from both sides
• Multiplying or dividing both sides by a positive or negative value carefully

Understanding these techniques allows us to make accurate statements about ranges of values that can satisfy given conditions.

Notation

Mathematical notation provides a concise way to represent mathematical concepts and processes. Interval notation is a way to express inequalities in a compact form using brackets and parentheses.

• Square brackets \([ \, or \, ]\) indicate that the endpoint is included in the interval.
• Parentheses \(( \, or \, )\) indicate that the endpoint is not included in the interval.

In our example, the inequality \(3 \leq x<8\) translates to \([3, 8)\) in interval notation.
This means 3 is part of the solution set, but 8 is not.
Using this notation helps in clearly and concisely describing the set of values that satisfy the inequality. It is especially useful in advanced mathematics, calculus, and when graphing solutions to inequalities.