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Chapter 2: Problem 45

For the following exercises, write the set in interval notation. $$ \\{x | x < 4\\} $$

Short Answer

Expert verified
\((-\infty, 4)\)

Step by step solution

01

Understand Set Notation

The set \( \{x | x < 4\} \) represents all real numbers \(x\) that are less than 4. The vertical bar '|' is read as 'such that'. So this set contains all numbers less than 4.
02

Identify the Interval Type

Since the set includes numbers less than 4 but not equal to 4, it is open on the right end. There are no lower bounds specified, so it extends infinitely in the negative direction. This means the interval is of the form \((-\infty, 4)\).
03

Write in Interval Notation

Based on step 2, express the set in interval notation: \((-\infty, 4)\). The parentheses indicate that 4 is not included in the set, which matches the original set condition \(x < 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Set Notation
Set notation is a way to specify a collection of elements, often numbers, that satisfy a certain condition. The set \( \{x \mid x < 4\} \) tells us about all elements, which in this case are the real numbers less than 4. The symbol \('|\) is crucial because it translates to "such that". In our example, it indicates that the set contains all the \(x\) which make the condition \(x < 4\) true. This kind of notation is efficient for describing complex groups of numbers and is widely used in mathematics.

Here's why set notation is helpful:
  • It clearly defines what is included or not included in the set.
  • It is concise and easy to write, especially with complex conditions.
  • It is flexible, allowing mathematicians to express different constraints simply.
Understanding Real Numbers
Real numbers are all the numbers you can think of on the number line. This includes positive numbers, negative numbers, whole numbers (like 0, 1, 2), fractions, decimals, and irrational numbers like \(\pi\). Real numbers are essentially all the numbers that you encounter in standard mathematical contexts, making them incredibly important in calculus and algebra.

They can be divided into two main categories:
  • Rational numbers: Numbers that can be expressed as a fraction, where both numerator and denominator are integers.
  • Irrational numbers: Numbers that cannot be expressed as fractions, such as \(\sqrt{2}\).
Understanding real numbers is critical since they form the foundation of many mathematical concepts. For instance, in the set \(\{x \mid x < 4\}\), the numbers less than 4 are real numbers, extending infinitely in the negative direction.
Understanding Inequalities
Inequalities express the relative size or order of two values. For instance, \(x < 4\) indicates that \(x\) is less than 4. Unlike equations, which show equality, inequalities show a range of possible solutions.

Here are some key components of inequalities:
  • <: Less than
  • >: Greater than
  • \( \leq \): Less than or equal to
  • \( \geq \): Greater than or equal to

In the exercise \( \{x \mid x < 4\} \), the inequality \(x < 4\) shows that \(x\) can be any real number less than 4. Consequently, there's no upper boundary like 4 included in the set; this leads to an open interval notation \(( -\infty, 4)\). Such representations are essential for specifying non-discrete answers, where a traditional equation is not sufficient.

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Most popular questions from this chapter

For the following exercises, solve the following polynomial equations by grouping and factoring. $$ m^{3}+m^{2}-m-1=0 $$

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-1|>2 $$

For the following exercises, write the interval in set-builder notation. $$ (4, \infty) $$

For the following exercises, solve the rational exponent equation. Use factoring where necessary. $$ x^{\frac{7}{3}}-3 x^{\frac{4}{3}}-4 x^{\frac{1}{3}}=0 $$

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x-4|<3 $$
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