Problem 35 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

If the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all cases, indicate the interval on the number line. $$ \text { (a) }-1

Step-by-Step Solution

Answer

The interval notation for the inequality '-1

Step 1: Understand the inequality

The given inequality is '-1

Step 2: Express the inequality in interval notation

The interval notation for 'x is greater than -1' is \((-1, +\infty)\). The parenthesis on '-1' indicates that -1 itself is not included in the interval.

Step 3: Represent the interval on the number line

To represent the interval \((-1, +\infty)\) on the number line, draw a line. Make a mark on this line at -1. Place a small open circle on -1 (to show that -1 is not included) and draw an arrow towards the right (to show that every number greater than -1 is included in the interval).

Key Concepts

Inequality

An inequality is a mathematical statement that compares two expressions. It tells us if one expression is greater than, less than, or possibly equal to another. For example, the inequality \(-1 < x\) suggests that the value of \(x\) is any number greater than \(-1\). In solving inequalities, we look for all possible values that can satisfy the given condition. This kind of solution is known as a solution set. Inequalities are fundamental in calculus and many other math areas because they show relationships between numbers in a range instead of giving a single value.

Number Line

A number line is a visual representation of numbers on a straight line. It allows us to see the relative positions and distances between numbers. When working with inequalities, the number line helps us visualize the solution set. To show \(-1 < x\), a mark is placed at -1, and an open circle is drawn over this mark. The open circle indicates that -1 itself isn't part of the solution. An arrow extending to the right represents all numbers greater than -1. This visual makes it easy for us to grasp which numbers are included in the inequality solution.

Interval Representation

Interval notation is a concise way to describe a set of numbers. It uses parentheses and brackets to indicate whether endpoints are included in the interval. For the inequality \(-1 < x\), we use the interval notation \((-1, +\infty)\). The parentheses around -1 indicate that -1 is not part of the interval. The plus infinity symbol \(+\infty\) simply indicates the set continues indefinitely in the positive direction.

• Parentheses \((a, b)\) mean \(a\) and \(b\) are not included.
• Brackets \([a, b]\) mean \(a\) and \(b\) are included.
• When using \(+\infty\) or \(-\infty\), always use parentheses since infinity is not a specific number.

Interval representation helps us simplify the description of complex solution sets.

Calculus

Calculus is a branch of mathematics focused on change, such as rates of change and accumulations. Inequalities play a critical role because they define the domains and ranges of functions, as well as constraints in optimization problems. Understanding how to express solutions in interval notation is crucial in calculus for defining where functions align with specific conditions. When analyzing functions, knowing how to work with inequalities helps you determine where a function is increasing or decreasing. This type of analysis is vital for sketching graphs or solving real-world problems. Therefore, mastering inequality solving, interval notation, and number lines lays the groundwork for success in calculus.

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Problem 34

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Problem 36

Other exercises in this chapter

Problem 33

For Problems 33 through 35, if the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it

View solution

Problem 34

If the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all

View solution

Problem 36

Let \(f(x)=2 x^{2}+x\). Find the following. (a) \(f(3)\) (b) \(f(2 x)\) (c) \(f(1+x)\) (d) \(f\left(\frac{1}{x}\right)\) (e) \(\frac{1}{f(x)}\)

View solution

Problem 37

A right circular cylinder is inscribed in a sphere of radius \(5 .\) (a) Express the volume of the cylinder as a function of its radius, \(r\). (b) Express the

View solution

Key Concepts

Inequality

An inequality is a mathematical statement that compares two expressions. It tells us if one expression is greater than, less than, or possibly equal to another. For example, the inequality \(-1 < x\) suggests that the value of \(x\) is any number greater than \(-1\). In solving inequalities, we look for all possible values that can satisfy the given condition. This kind of solution is known as a solution set. Inequalities are fundamental in calculus and many other math areas because they show relationships between numbers in a range instead of giving a single value.

Number Line

A number line is a visual representation of numbers on a straight line. It allows us to see the relative positions and distances between numbers. When working with inequalities, the number line helps us visualize the solution set. To show \(-1 < x\), a mark is placed at -1, and an open circle is drawn over this mark. The open circle indicates that -1 itself isn't part of the solution. An arrow extending to the right represents all numbers greater than -1. This visual makes it easy for us to grasp which numbers are included in the inequality solution.

Interval Representation

Interval notation is a concise way to describe a set of numbers. It uses parentheses and brackets to indicate whether endpoints are included in the interval. For the inequality \(-1 < x\), we use the interval notation \((-1, +\infty)\). The parentheses around -1 indicate that -1 is not part of the interval. The plus infinity symbol \(+\infty\) simply indicates the set continues indefinitely in the positive direction.

• Parentheses \((a, b)\) mean \(a\) and \(b\) are not included.
• Brackets \([a, b]\) mean \(a\) and \(b\) are included.
• When using \(+\infty\) or \(-\infty\), always use parentheses since infinity is not a specific number.

Interval representation helps us simplify the description of complex solution sets.

Calculus

Calculus is a branch of mathematics focused on change, such as rates of change and accumulations. Inequalities play a critical role because they define the domains and ranges of functions, as well as constraints in optimization problems. Understanding how to express solutions in interval notation is crucial in calculus for defining where functions align with specific conditions. When analyzing functions, knowing how to work with inequalities helps you determine where a function is increasing or decreasing. This type of analysis is vital for sketching graphs or solving real-world problems. Therefore, mastering inequality solving, interval notation, and number lines lays the groundwork for success in calculus.