Problem 22 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ -34 \leq x \leq 12 $$

Step-by-Step Solution

Answer

The solution is \([-34, 12]\) and is represented by a line segment between -34 and 12 with solid endpoints on the number line.

Step 1: Understand the Inequality

The inequality given is \(-34 \leq x \leq 12\). This means that \(x\) is a number greater than or equal to \(-34\) and less than or equal to \(12\). We need to represent all the values of \(x\) that satisfy this inequality on a number line.

Step 2: Draw the Number Line

Draw a horizontal line to represent the number line. Mark points on this line at \(-34\) and \(12\). These values are the boundaries for our solution set.

Step 3: Shade the Interval on the Number Line

Since the inequality is inclusive (as indicated by the "\(\leq\)") for both ends, draw a solid dot at \(-34\) and another solid dot at \(12\). Then, shade the region between these two points to show all numbers \(x\) such that \(-34 \leq x \leq 12\).

Step 4: Write Interval Notation

The interval notation for the solution set starts with the smallest number, \(-34\), and ends with the largest number, \(12\). Since both endpoints are included, their brackets are square brackets. Thus, the interval notation is \([-34, 12]\).

Key Concepts

Understanding Inequalities

Inequalities are a powerful way to express the range of possible values for a variable. In this case, the inequality \(-34 \leq x \leq 12\) tells us that \(x\) is a number that can be equal to or lie between \(-34\) and \(12\). The symbol "\(\leq\)" means "less than or equal to," so both boundaries are included in the solution set.
In this inequality:

• \(-34\) is the lower boundary. \(x\) can be equal to \(-34\) or any number greater than it.
• \(12\) is the upper boundary. \(x\) can be equal to \(12\) or any number less than it.

When working with inequalities, it is crucial to pay attention to the symbols used, as they determine whether the boundary points are part of the solution (inclusive, \(\leq\)) or not (exclusive, \(<\)).
This inequality is inclusive at both ends, meaning the solution set includes the numbers \(-34\) and \(12\). Understanding this concept is fundamental when graphing such inequalities and translating them to interval notation.

Interval Notation

Interval notation is a concise way to describe the set of solutions for an inequality on a number line. For the inequality \(-34 \leq x \leq 12\), interval notation captures the complete range of values \(x\) can take.
Here's how it works:

• Begin with the smallest number in the range, \(-34\).
• End with the largest number in the range, \(12\).
• Use square brackets \([ \text{ and } ]\) to include endpoints when the inequality is inclusive.

For our example, both endpoints are included because of the "\(\leq\)" in the inequality. Thus, the interval notation is \([-34, 12]\). This notation communicates easily that any number \(x\) between and including \(-34\) and \(12\) is a solution.
Interval notation is often preferred because it clearly shows the boundaries and inclusivity in a simple, standardized form.

Graphical Representation

Creating a graphical representation on a number line helps in visualizing the solution to an inequality. By drawing a line and shading the appropriate region, you can have a quick, easy-to-understand picture of all possible solutions.
To graph the solution to the inequality \(-34 \leq x \leq 12\) on a number line:

• First, draw a straight horizontal line to represent the number line.
• Mark two points on the line corresponding to the values \(-34\) and \(12\). These points indicate the boundaries of the solution set.
• Place solid dots at \(-34\) and \(12\) to show that these endpoints are part of the solution (because the inequality is inclusive).
• Shade or draw a thick line between these two points to indicate that all numbers from \(-34\) to \(12\) are included within the solution set.

This graphical representation provides a clear and immediate picture of the inequality's solutions, allowing you to see directly which \(x\) values are included. Graphing is especially helpful when communicating and verifying solutions to others.