Problem 12 - Elementary Algebra

Question

Graph all solutions on a number line and provide the corresponding interval notation. $$ x<34 $$

Step-by-Step Solution

Answer

On a number line: open circle at 34, shaded left; Interval notation: \((-\infty, 34)\).

Step 1: Understand the Inequality

The problem gives us the inequality \(x < 34\). This inequality includes all real numbers less than 34.

Step 2: Identify the Solution Set

The solution set contains all numbers to the left of \(34\) on a number line, since \(x < 34\) means \(x\) is any number less than \(34\).

Step 3: Graph the Solution on a Number Line

First, draw a horizontal line to represent the number line. Mark a point at \(34\). Use an open circle at \(34\) to indicate that \(34\) itself is not included in the solution set. Shade the line extending to the left from \(34\) to show all numbers less than \(34\).

Step 4: Write the Interval Notation

The interval notation for the solution is \((-\infty, 34)\). The parenthesis indicates \(34\) is not included, and we use negative infinity \((-\infty)\) to show that the solution goes infinitely to the left.

Key Concepts

Understanding the Number Line

The number line is a straight, horizontal line that represents all real numbers in a sequential manner. Numbers increase as you move from left to right. The number line for the inequality \(x < 34\) helps visualize which values satisfy the condition.
To graph \(x < 34\) on the number line:

• Locate the point at 34. This point is crucial as it acts as a boundary for permissible values.
• Draw an open circle on the point at 34, indicating the number 34 is not included in the solution.
• Shade the segment of the number line that stretches left from 34. This shaded section represents all numbers that are less than 34.

The open circle signifies that the point (34 in this case) is not part of the solution, essentially showing that \(x\) should always be lesser than 34 and not equal to it.

Using Interval Notation

Interval notation is a concise way to describe sets of numbers that represent solutions to an inequality. It uses parentheses and brackets to show whether endpoints are included in the set.
For the inequality \(x < 34\):

• Because \(34\) is not included, we use a parenthesis at 34, written as \(34\).
• Since the values continue indefinitely to the left, we use negative infinity, \(-\infty\), indicated by \((-\infty, 34)\).

Parentheses are used next to infinity symbols because infinity itself is a concept, not a specific number that can be reached or included in a set. This notation clearly communicates that \(x\) can be any number less than 34.

Graphical Representation of Inequalities

Graphical representation is a powerful tool that gives a visual touch to mathematical concepts like inequalities. By plotting inequalities on a number line, you make it easier to interpret the set of solutions.
Here's how the graphical process works:

• First, identify the boundary, which is the number 34 for \(x < 34\). Nevertheless, 34 isn't included in the solution set, marked by an open circle.
• The line extending to the left represents all possible values for \(x\), visually showing the concept of having an infinite number of solutions within a certain range.

This graphing process not only helps in understanding the given inequality but also assists in grasping similar problems encountered in mathematics. Emphasizing visualization accelerates comprehension and retention of such concepts.