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Chapter 2: Problem 44

For the following exercises, write the set in interval notation. $$ \\{x | x \geq 7\\} $$

Short Answer

Expert verified
The interval notation for \( \{ x | x \geq 7 \} \) is \([7, \infty)\).

Step by step solution

01

Understanding the Set Notation

The given set notation \( \{ x | x \geq 7 \} \) describes a set of all real numbers \( x \) such that \( x \) is greater than or equal to 7. In this notation, \( \geq \) indicates that 7 is included in the set.
02

Converting to Interval Notation

In interval notation, we use brackets to indicate whether endpoints are included or not. A square bracket "\([\)" indicates inclusion, and a parenthesis "\()\)" indicates exclusion. Since the expression \( x \geq 7 \) includes 7 and extends to infinity, we write the interval as \([7, \infty)\).
03

Understanding Infinity in Interval Notation

The symbol \( \infty \) means that the interval extends indefinitely. Since infinity is not a real number, we always use a parenthesis "\()\)" with infinity. Thus, the interval \([7, \infty)\) means all numbers from 7 to infinity, including 7.

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Key Concepts

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

Set Notation
When we see a set written in set notation, what we're looking at is a description of a group of numbers that share a specific property. For example, the set \( \{ x | x \geq 7 \} \) tells us about every number \( x \) that meets the criteria \( x \geq 7 \). We use the curly braces \( \{ \} \) to enclose the set, and the vertical bar \( | \) to mean "such that." As a result, this set includes all real numbers that are greater than or equal to 7:
  • \( x \) includes numbers like 7, 8, 10, and 100, but not 6.9
  • The inequality \( \geq \) specifies that 7 is part of the set
When converting to interval notation, this same set will be represented using a different system of symbols.
Real Numbers
Real numbers are the backbone of most of the numbers we use every day. They include all the numbers you can find on the number line:
  • Whole numbers (like -3, 0, and 7)
  • Fractions (like \( \frac{1}{2} \) and -\( \frac{3}{4} \))
  • Decimals (like 3.14 or -2.718)
In the context of our set notation \( \{ x | x \geq 7 \} \), when we say real numbers, we're highlighting that every point from 7 onwards, without skipping any decimal or fraction, is included. Real numbers form a continuous line, and our set captures every part of that line starting from 7.
Inequalities
Inequalities are used in mathematics to describe the relative size or order of two objects. They are symbols like \(<, \leq, \geq, eq, \) and \(>\):
  • \(\geq\) means "greater than or equal to," which includes the value itself and anything larger.
  • \(\leq\) stands for "less than or equal to," indicating inclusion of a value and everything smaller.
In the set \( \{ x | x \geq 7 \} \), the inequality \( \geq \) tells us that 7 is the smallest number that can be part of this set, and every number greater than 7 is also included. It's an inclusive measure, meaning the endpoint is part of the set.
Infinity
Infinity is a concept rather than a number and is used to express unboundedness or endlessness in mathematics. In interval notation, we use infinity to indicate that the set of numbers extends indefinitely:
  • Infinity is denoted by the symbol \( \infty \).
  • When representing intervals like \([7, \infty)\), the \( \infty \) suggests the numbers continue forever.
Crucially, because infinity isn't a number you can "reach" or include on a number line, we always use a parenthesis \( ) \) rather than a bracket \( ] \) with it. This is true whether the interval is approaching positive infinity or negative infinity.

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Most popular questions from this chapter

For the following exercises, write the set in interval notation. $$ \\{x | x \text { is all real numbers }\\} $$

For the following exercises, solve the rational exponent equation. Use factoring where necessary. $$ x^{\frac{2}{3}}=16 $$

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. $$ |x+7| \leq 4 $$

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) 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, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st 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. $$ |x+2| \geq 5 $$

For the following exercises, describe all the \(x\) -values within or including a distance of the given values. Distance of 5 units from the number 7
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