Problem 20 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ -30

Step-by-Step Solution

Answer

Graph: open circles on \(-30\) and \(0\), shaded between; Interval notation: \((-30, 0)\).

Step 1: Understand the Inequality

The given inequality is \(-30 < x < 0\). This means that \(x\) is any number greater than \(-30\) and less than \(0\). There are no equalities, so \(x\) cannot be exactly \(-30\) or \(0\). It also implies that any number between \(-30\) and \(0\) is a solution of this inequality.

Step 2: Graph the Inequality on a Number Line

To graph \(-30 < x < 0\) on a number line, you will draw a number line with numbers marked including \(-30\) and \(0\). Since \(x\) cannot be equal to \(-30\) or \(0\), we represent these values with open circles on the number line to indicate that these endpoints are not included in the solution set. Shade the region between the open circles.

Step 3: Determine the Interval Notation

In interval notation, use parentheses to indicate that the endpoints are not included. For the inequality \(-30 < x < 0\), the interval notation equivalent is \((-30, 0)\). This shows that \(x\) is between \(-30\) and \(0\), not including \(-30\) and \(0\) themselves.

Key Concepts

Number Line

A number line is a visual representation of numbers in a straight line.
It's a useful tool to illustrate inequalities like \(-30 < x < 0\). On a number line:

• Draw a horizontal line.
• Mark the points -30 and 0.
• These points serve as boundaries of the inequality.

Once marked, the solution set of the inequality \(-30 < x < 0\) is shown. This is done by shading the region between -30 and 0.
Doing this helps visually identify all possible solutions as any number between them on the line.

Interval Notation

Interval notation offers a concise way of representing a range of solutions.
For the inequality \(-30 < x < 0\), we write:

• \((-30, 0)\)

The use of parentheses \(()\) indicates that the endpoints -30 and 0 are not included.
Parentheses represent open intervals, meaning values close to but not equal to the boundaries are part of the solution. This notation is efficient and widely used in mathematics to describe ranges succinctly.

Open Circles

In graphing inequalities, open circles signify that endpoints are not part of the solution.
For \(-30 < x < 0\), both -30 and 0 will be represented by open circles on the number line.
Here's why:

• Open circles are used instead of filled circles or dots.
• This indicates that the number itself isn't included in the solution.
• In contrast, a closed or filled circle would mean inclusion.

Using open circles helps emphasize the strict inequality, visually ensuring clarity in which numbers are part of the solution range.