Problem 41 - Algebra and Trigonometry

Question

\(41-46=\) Express the interval in terms of inequalities, and then graph the interval. $$ (-3,0) $$

Step-by-Step Solution

Answer

The interval \((-3, 0)\) is expressed as \(-3 < x < 0\), and is graphed with open circles at -3 and 0.

Step 1: Understand the Notation of the Interval

The interval \((-3, 0)\) is an open interval, meaning it does not include the endpoints -3 and 0. It represents all real numbers \(x\) such that \(-3 < x < 0\).

Step 2: Express the Interval as an Inequality

To express the interval \((-3, 0)\) in terms of inequalities, we write it as: \(-3 < x < 0\). This shows that \(x\) can take any value greater than -3 and less than 0.

Step 3: Graph the Interval on a Number Line

To graph this interval on a number line, draw a horizontal line and mark the numbers -3 and 0 on it. Place open circles (indicating that these points are not included) at -3 and 0. Shade the line in between these circles to indicate all numbers that lie within the interval \(-3 < x < 0\).

Key Concepts

Open Intervals

An open interval represents a range of numbers that do not include its endpoints. For instance, in the interval \((-3, 0)\), the numbers -3 and 0 are not part of the interval. This means any number within this range, like -2 or -1.5, is included, but -3 and 0 are not. Open intervals are commonly used in mathematics to describe situations where a boundary is not included.
Understanding open intervals can be crucial in various branches of mathematics, especially in calculus and real analysis. They allow for the description of continuous ranges, omitting the strict boundaries. Remember:

• Round brackets ( ) are used for open intervals.
• Endpoints are not included.
• Examples: \((-1, 5)\), \((2, 7)\).

Inequalities

Inequalities are mathematical expressions that indicate that one quantity is larger or smaller than another. In our case, expressing the open interval \((-3, 0)\) using inequalities translates to \(-3 < x < 0\). This tells us that \(x\) can be any value greater than -3 and less than 0.
Working with inequalities helps identify a range of potential solutions, rather than a single answer. They are pivotal in solving equations, optimizing problems, and much more. Key points to remember when dealing with inequalities include:

• "<" and ">" are strictly less than and greater than, indicating open ends.
• "<=" and ">=" include the endpoint, indicating closed ends (not applicable in this open interval).
• Reversing the inequality sign is done when multiplying or dividing by a negative number.

Number Line Graphing

Graphing on a number line visualizes real number intervals. For the open interval \((-3, 0)\), we need to show the range of numbers excluding -3 and 0. To do this:
- Draw a horizontal number line.- Mark the points -3 and 0 on the line.- Use open circles at -3 and 0, indicating those endpoints are not included.- Shade or draw a line segment between the open circles to show all numbers between -3 and 0.
Number line graphing is a helpful tool for understanding how intervals work and how they relate to inequalities. It visually demonstrates the solution to an inequality, offering insight into the concepts of bounded and unbounded intervals.

In practice, number line graphs aid greatly in both simple arithmetic exercises and more complex problem-solving situations.