Problem 53 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$(-\infty, \pi]$$

Step-by-Step Solution

Answer

The inequality is \(x \leq \pi\) and all values left of and including \(\pi\) are on the number line.

Step 1: Identify the Interval

The interval given is \((-\infty, \pi]\). This means that the interval starts from negative infinity and goes up to and includes \(\pi\). In interval notation, the round bracket \((-\infty,\) indicates that negative infinity is not included and the square bracket \(\pi]\) indicates that \(\pi\) is included.

Step 2: Express in Inequality Notation

To express the interval \((-\infty, \pi]\) in inequality notation, use the variable \(x\) to represent the numbers in the interval. The inequality for this interval is \(x \leq \pi\), meaning all numbers \(x\) that are less than or equal to \(\pi\) are part of the interval.

Step 3: Represent on a Number Line

The number line representation will start from the far left, representing negative infinity, and extend to \(\pi\). At \(\pi\), draw a solid dot to indicate that \(\pi\) is included in the interval. Shade the line to the left of \(\pi\) to indicate all numbers less than \(\pi\) are part of the interval.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real numbers. It is used to describe a set of numbers from the "left endpoint" to the "right endpoint." In this system:

• Brackets and parentheses are used to indicate whether endpoints are included or excluded.
• Square brackets, like \([ \text{ or } ]\), mean that the endpoint is included in the interval.
• Parentheses, like \(( \text{ or } )\), mean that the endpoint is not included.
• For example, the interval \((-\infty, \, \pi] \) starts from negative infinity and goes up to and **includes** \( \pi \).

Interval notation is handy because it neatly summarizes the range of numbers we're dealing with.
For instance, \( (-\infty, \, \pi] \) suggests a continuous range of all numbers less than or equal to \( \pi \), which is particularly useful in calculus and algebra.

Number Line Representation

Number line representation provides a visual way to show the set of real numbers that satisfies an inequality or an interval. Visual depictions like number lines can help make connections between algebraic expressions and their real-world applications.With intervals:

• An open circle indicates that an endpoint is not included.
• A closed or filled circle is used if the endpoint is included.
• The line is shaded to show all numbers in the interval.

For the interval \((-\infty, \, \pi]\):
- Place a solid dot at \(\pi\) because it is included.
- Shade all the way to the left towards negative infinity to represent all numbers less than \(\pi\).
This representation makes it easy to see everything included in the interval at a glance.
Understanding number line representations is a foundational skill in graphing and analyzing inequalities.

Inequalities in Mathematics

Inequalities are mathematical expressions involving symbols like \( <, \leq, >, \text{and } \geq \). They express a range of possible values rather than a precise number, which is key in scenarios where exact values aren't needed or don't exist.Common inequality symbols include:

• \( < \) for "less than"
• \( \leq \) for "less than or equal to"
• \( > \) for "greater than"
• \( \geq \) for "greater than or equal to"

In this example, \( x \leq \pi \) is the inequality notation for the interval \((-\infty, \pi]\).
This inequality tells us that \(x\) can be any number that is less than or equal to \(\pi\).
Inequalities are very practical in problem-solving, providing a way to deal with ranges in mathematics.
They allow you to express relationships between quantities that are not confined to a single value, expanding mathematical problem-solving beyond the realm of equations.
Understanding how to translate between different forms of inequality expressions is crucial for tackling problems across various branches of mathematics.