Problem 46 - Contemporary Precalculus

Question

Use interval notation to denote the set of all real numbers \(x\) that satisfy the given inequality. $$7

Step-by-Step Solution

Answer

Question: Find the set of all real numbers x that satisfy the inequality \(7 < x < 77\) and express the range using interval notation. Answer: The set of all real numbers x that satisfy the inequality is represented by the interval notation (7, 77).

Step 1: Use Inequality Signs

We are trying to find the set of all real numbers x that satisfy \(7 < x < 77\). We'll use the inequality signs to represent the range using interval notation.

Step 2: Write the Interval Notation

In interval notation, we use open brackets "(" to denote an exclusive boundary, meaning the number itself is not included, and closed brackets "[" to denote an inclusive boundary, meaning the number itself is included in the interval. Here, since 7 < x, number 7 is not part of the interval; thus, we use an open bracket. Similarly, 77 is also not part of the interval; so we use an open bracket again. The correct interval notation for this inequality is: $$(7, 77)$$ This interval denotes the range of all real numbers between 7 and 77, exclusive of both 7 and 77.

Key Concepts

Understanding Inequality

In mathematics, inequalities express the relationship between two values, showing that one is greater than, less than, or potentially equal to the other. For example, the inequality \(7 < x < 77\), means that the value of \(x\) lies between 7 and 77 but does not include these boundary values.
Inequalities are often used to define ranges for unknown values and play a crucial role in algebra and calculus. They can be expressed using symbols like:

• < - less than
• > - greater than
• ≤ - less than or equal to
• ≥ - greater than or equal to

These symbols help succinctly summarize constraints or conditions. By mastering inequalities, students can analyze mathematical expressions more fluidly and solve a wide range of real-world and theoretical problems.

Working with Real Numbers

Real numbers encompass all numbers on the continuous number line, including both rational numbers like fractions and integers, and irrational numbers such as \(\pi\) and \(\sqrt{2}\).
When dealing with inequalities involving real numbers, the solution set typically includes all values within a specified range. An inequality like \(7 < x < 77\) identifies a set of real numbers greater than 7 but less than 77, excluding these boundaries.
Real numbers are incredibly important in mathematics because they provide a complete, consistent framework for measuring quantities and analyzing mathematical models. They allow us to perform arithmetic operations and understand more complex structures like limits, continuity, and derivatives.

The Role of Mathematics Education

Mathematics education aims to equip learners with the skills needed to understand and solve mathematical problems. This includes understanding concepts such as inequalities and real numbers.
Through education, learners develop critical thinking and problem-solving skills that are applicable in many fields. The importance of exercises like working with interval notation lies in their ability to strengthen comprehension of numerical relationships and algebraic structures.
Good mathematics education encourages a step-by-step approach, as seen in the solution using interval notation, to facilitate understanding from foundational basics to more complex applications. By comprehensively addressing these core concepts, students are better prepared for academic challenges and real-world tasks.