Problem 21 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ -58

Step-by-Step Solution

Answer

Interval notation: \((-58, 18)\); graph: open circles at \(-58\) and \(18\), shading in between on the number line.

Step 1: Understand the Inequality

The given inequality is \(-58 < x < 18\). This means that \(x\) is any number greater than \(-58\) and less than \(18\).

Step 2: Identify the Interval

To express this solution in interval notation, identify the lower and upper bounds of \(x\). The lower bound is \(-58\) and the upper bound is \(18\). Since \(x\) cannot equal \(-58\) or \(18\), we use parentheses in the interval notation. Thus, the interval in notation form is \((-58, 18)\).

Step 3: Graph the Interval on a Number Line

To graph the interval, use a number line and mark the points \(-58\) and \(18\). Indicate that \(-58\) and \(18\) are not included in the interval with open circles on the number line at these points. Then, shade the region between \(-58\) and \(18\) to show all possible values of \(x\). The open circles at both ends indicate that the endpoints \(-58\) and \(18\) are not included in the solution set.

Key Concepts

Understanding the Number Line

The number line is a visual representation used in mathematics to show real numbers in a straight line. Think of it as a ruler stretching infinitely in both directions. It's a fundamental tool in elementary algebra for understanding inequalities, like the one presented:

\(-58 < x < 18\).

Here’s how to use a number line for such inequalities:

• First, draw a straight line and pick a point to represent zero.
• Mark numbers along the line according to their value. In this exercise, focus on two key numbers: -58 and 18.
• Since these numbers are boundaries where \(x\) is not included, use open circles on the line at -58 and 18.
• Shade the area between these points to indicate all values \(x\) could take.

This shading shows the continuous range of numbers between -58 and 18 that satisfy the inequality.

Demystifying Interval Notation

Interval notation is a concise way to describe a range of numbers. It's especially useful in expressing solutions to inequalities. In the inequality \(-58 < x < 18\), we can use interval notation to express the values that \(x\) can take:

• The interval begins just after -58 and ends just before 18.
• Since both endpoints are not included, parentheses are used: \((-58, 18)\).

Think of interval notation like a shorthand. It tells us everything we need to know about where \(x\) lies:

• Round brackets or parentheses \(()\) indicate that the endpoints are not included (known as open intervals).
• Square brackets \([]\) indicate that the endpoints are included, but in this case, they are not used.

Interval notation provides a precise way to communicate mathematical concepts without lengthy explanations.

Basics of Elementary Algebra

Elementary algebra introduces students to the core concepts of manipulating mathematical expressions and solving equations. It's like the alphabet of mathematics, laying the foundation for advanced topics. An essential part of this is understanding inequalities, such as the inequality in our example:

• Inequalities show the relationship between two expressions or numbers, indicating that one is larger or smaller.
• Solving inequalities often involves similar steps to solving equations but requires careful attention to the direction of the inequality.

In this exercise, the aim is to identify all possible values of \(x\) that satisfy \(-58 < x < 18\):

• Recognize that inequalities can have multiple solutions. Here, every number between -58 and 18 is a potential solution.
• Using algebraic properties and operations is crucial to isolate and solve for \(x\) in more complex inequalities.

By using these principles in algebra, you can easily decipher and represent inequalities with number lines and interval notations.