Problem 7 - Algebra for College Students

Question

Express the given inequality in interval notation and sketch a graph of the interval. \(x \leq 2\)

Step-by-Step Solution

Answer

The interval is \((-\infty, 2]\) and the graph includes a solid dot at 2 with a leftward arrow.

Step 1: Identify the Inequality

The inequality given is \(x \leq 2\). This means that all values of \(x\) are less than or equal to 2. To express this in interval notation, we determine the lower and upper bounds of the interval.

Step 2: Express in Interval Notation

Since the inequality \(x \leq 2\) accepts all values less than or equal to 2, it means \(-\infty < x \leq 2\). Hence, in interval notation, this inequality is expressed as \( (-\infty, 2] \). The parenthesis \((-\infty, 2]\) indicates that the interval includes all numbers up to and including 2, but not beyond.

Step 3: Sketch the Graph

On a number line, draw a horizontal line. Place a solid dot or a filled-in circle at the point corresponding to 2 to show inclusion. Then, draw a line with an arrow extending to the left of 2 to indicate all numbers less than 2 are included in the interval. The arrow represents that the values continue infinitely in the negative direction.

Key Concepts

Inequality

An inequality is like a comparison between two quantities, showing whether one is smaller, larger, or perhaps equal to the other. In our given exercise, the inequality is expressed as \(x \leq 2\). Here's what each part of this symbol means:

• The symbol \(\leq\) stands for 'less than or equal to'. This means we're looking at values of \(x\) that aren't larger than 2.
• The number 2 is the specific value that \(x\) can be equal to, and it functions as our upper limit.

Inequalities are a fundamental part of mathematics that help us describe sets of numbers. They're especially useful in many real-world scenarios, like determining maximum capacities or limiting ranges. In essence, an inequality like \(x \leq 2\) creates a boundary and tells us about everything beyond that boundary up to a certain point.

Number Line Graph

A number line graph is a simple visual representation that helps us understand inequalities more clearly. Drawing this graph allows you to see the range of numbers that the inequality encompasses. Here's how it works for our inequality \(x \leq 2\):

• First, draw a horizontal line. This represents all the possible values.
• Locate the number 2 on the line as this is our boundary point.
• Place a filled-in circle at 2 since the inequality includes this value ("less than or equal to").
• Draw an arrow extending to the left of 2. This arrow shows that all numbers to the left (meaning less than 2) are also part of the solution.

Number line graphs are powerful tools since a quick glance can help you see both the boundary and the direction in which the values are included. They answer "where" and "what" visually about any inequality solution.

Mathematical Notation

Mathematical notation is the language of math. It uses symbols, rather than words, to convey ideas in a compact and universal manner. The transition from inequality to interval notation is a perfect example of this language. Let's break down our exercise of converting \(x \leq 2\) into interval notation:

• The inequality \(x \leq 2\) is written using the crisp interval notation \((-\infty, 2]\).
• The symbol \((\-\infty\) signifies that there is no lower limit to our values —they stretch infinitely into the negative.
• A square bracket \([2]\) is used next to 2 to indicate that 2 is included inside the set.

This notation is concise but might look intimidating at first. Once you understand each part's function, you'll see how it neatly encompasses the idea of inequality. Whether in equations, graphs, or real-world constraints, mathematical notation serves as a bridge translating broad concepts into refined expressions.