Problem 34 - 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. $$ \text { (a) }-7 \leq x<-5 $$

Step-by-Step Solution

Answer

The interval notation for \(-7 \leq x<-5\) is \([-7, -5)\).

Step 1: Understand the Inequality

The inequality is \(-7 \leq x<-5\). This inequality shows that x is greater than or equal to -7 (included in the set) and less than -5 (not included in the set).

Step 2: Convert Inequality to Interval Notation

Given the inequality \(-7 \leq x<-5\), it can be written in interval notation as \([-7, -5)\). The square bracket around -7 indicates that -7 is included in the interval (because of the 'less than or equal to' symbol), and the round bracket around -5 indicates that -5 is not included in the interval (due to the 'less than' symbol).

Step 3: Represent the interval on Number Line

On the number line, represent the interval by marking a solid dot at -7 (because it's included in the set). Then, draw a line extending to -5 and mark an open circle at -5 (because it's not included in the interval).

Key Concepts

Understanding Inequalities

In the realm of mathematics, inequalities are statements that describe the relative size or order of two values. They play a crucial role in depicting ranges of values that a variable can take.

For example, the inequality \( -7 \leq x < -5 \) comprises two important parts: the 'greater than or equal to' sign (\( \geq \)) and the 'less than' sign (\( < \)). The first part, \( -7 \leq x \) states that \( x \) is greater than or equal to -7, while the second part, \( x < -5 \) tells us that \( x \) is strictly less than -5.

To interpret this, imagine a number line where every point represents a potential value for \( x \). The solutions to our inequality are all the points that lie between -7 and -5, including -7 but not including -5. It's critical to understand that the inequality symbol defines whether the endpoint is included in the solution set (closed dot) or not (open dot).

Number Line Representation

The number line representation is a visual tool that aids in the comprehension of inequalities and intervals. A number line is a straight, horizontal line with numbers placed at intervals according to their value.

When illustrating the interval from the inequality \( -7 \leq x < -5 \) on the number line, we start by placing a darkened dot at -7 to indicate that this number is part of the solution set. This is followed by a line extending to the right, towards -5, reflecting all the possible values \( x \) can take. As -5 is not included, we mark it with an open circle. This visual representation is powerful because it provides an immediate sense of the range and values within an interval.

Key Tips for Number Line Drawing

• Use a darkened dot for included endpoints.
• Use an open circle for excluded endpoints.
• Draw a continuous line between the endpoints to signify all the numbers in between.

Set Notation and Interval Notation

Students often encounter set notation when dealing with groups of numbers that belong to a particular category, based on certain conditions. With inequality \( -7 \leq x < -5 \) as our reference, the set notation would explicitly list the conditions that values of \( x \) must satisfy to be included in the set.

On the other hand, interval notation is a more compact form of set notation. It conveys the same information in a simpler way. For our given inequality, the interval notation is \([-7, -5)\). In this format, a closed bracket [ indicates inclusion of the endpoint (-7), and the open bracket ) signals the exclusion of the endpoint (-5).

Remember, interval notation is an efficient way to communicate information about sets without writing out lengthy verbal or symbolic descriptions. It's crucial to recognize and correctly interpret the brackets to understand the intervals they represent.