Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show the interval on a number line. $$ [-1,4) $$

Step-by-Step Solution

Answer

Draw a number line with labels, mark a closed circle at -1 and an open circle at 4, then shade the region between these points. The representation is: ``` ------------●¯¯¯¯¯¯¯¯¯○------------ -1 4 ```

Step 1: Draw a number line.

Draw a straight horizontal line, and label the numbers along the line. Make sure to include -1 and 4 on the number line.

Step 2: Mark the interval.

Locate -1 and 4 on the number line. Since -1 is included in the interval, draw a closed circle at -1. As 4 is not included in the interval, draw an open circle at 4.

Step 3: Shade the interval.

Shade the area on the number line that lies between the two marked points (closed and open circles), including the closed circle and not including the open circle. Our number line with the interval represented should look like this: ``` ------------●¯¯¯¯¯¯¯¯¯○------------ -1 4 ```

Key Concepts

Interval Notation

Interval notation is a way of describing a set of numbers along a number line. It uses parentheses and brackets to show which numbers are included in the set. For example,

• Brackets, like "[ ]", denote that an endpoint is included in the interval. This is known as "closed."
• Parentheses, like ")(", mean the endpoint is not included, referred to as "open."

In the interval \([-1, 4)\):

• The "-1" is included, so it gets a bracket.
• The "4" is not included, so it gets a parenthesis.

This notation helps to simplify the representation of ranges, especially when dealing with inequalities. Rather than writing "\(-1 \leq x < 4\)," interval notation compactly expresses the same idea. It's a valuable tool in algebra and calculus, aiding in the graphing and analysis of functions.

Number Line

A number line is a visual representation of numbers on a straight line. This tool is incredibly helpful when you're trying to understand different intervals or sets of numbers. When drawing a number line:

• Draw a horizontal line and evenly distribute numbers on it.
• Label important points according to the problem, like -1 and 4 in our exercise.

Number lines work well with interval notations; they provide a clear way to visualize which numbers are part of an interval.To represent the interval \([-1, 4)\):

• Locate -1 on the line and use a closed circle because it is included.
• Locate 4 and use an open circle because it is not included.
• Shade the area between -1 and 4 to represent all numbers in between.

This visual method helps in recognizing patterns and understanding how different sets interact.

Closed and Open Intervals

Closed and open intervals are critical concepts in mathematics that define the range of values included on a number line.An interval refers to the distance between two numbers, and it tells us which values are included in a set:

• A closed interval, represented with brackets \([a, b]\), means both endpoints are included. For instance, \([2, 5]\) includes 2 and 5 as well as every number in between.
• An open interval, represented with parentheses \((a, b)\), excludes both endpoints. \((2, 5)\) would include everything between 2 and 5, but not 2 and 5 themselves.

Sometimes, intervals can be a mix of both:

• The interval \([-1, 4)\), includes -1 (closed circle) but excludes 4 (open circle). This tells us -1 is part of the solution set, but 4 is not.

Understanding these distinctions is important for tasks such as integration, solving inequalities, and graphing functions.