Problem 53 - Precalculus Mathematics for Calculus

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$-2

Step-by-Step Solution

Answer

Interval notation: \((-2, 1]\).

Step 1: Understanding the Inequality

Firstly, let's interpret the inequality \(-2 < x \leq 1\). This means that \(x\) can take any value greater than \(-2\) but less than or equal to \(1\).

Step 2: Expressing in Interval Notation

In interval notation, values that are **not included** use parentheses \(()\) and values that **are included** use brackets \([]\). Here, \(-2\) is not included, so we use \((-2\), and \(1\) is included, so we use \(1]\). Thus, the interval notation is \((-2, 1]\).

Step 3: Graphing the Interval

To graph the interval, draw a number line. Plot \(-2\) and \(1\). At \(-2\), draw an open circle to denote that \(-2\) is not included in the interval. At \(1\), draw a closed circle to denote inclusion of \(1\). Shade the region on the number line between \(-2\) and \(1\) to represent all values of \(x\) within this interval.

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements that compare two values, showing that one is either less than, greater than, not equal to, or equal to. In the case of \(-2 < x \leq 1\), we are dealing with a compound inequality. This means that the inequality provides a range of acceptable values for \(x\):

• \(-2 < x\) indicates that \(x\) is greater than \(-2\), but \(-2\) itself is not included as a possible value for \(x\).
• \(x \leq 1\) implies that \(x\) can be exactly \(1\) or any value less than \(1\), including all numbers in between \(-2\) and \(1\).

Visualizing inequalities helps us understand which numbers fit within the defined range and keeps things organised when tackling more complex problems.

Exploring the Number Line

A number line is a simple yet powerful tool to visualize mathematical concepts like inequalities. It is a horizontal line with numbers placed at intervals. Key numbers are marked, which helps us see relationships between values at a glance.
When you want to represent an inequality, like \(-2 < x \leq 1\), the number line can show which numbers x can take:

• We start by marking our critical points, which in this case are \(-2\) and \(1\).
• An open circle is drawn at \(-2\) to indicate it is not part of the solution set.
• At \(1\), a closed circle is drawn because it is included within the set of possible \(x\) values.
• Finally, shading the section between these two points shows all numbers in the range \(-2 < x \leq 1\).

The number line provides a clear, visual representation of where the inequality holds true.

Graphing Intervals and Their Notation

Graphing intervals on a number line allows us to represent solutions to inequalities visually. This practice is crucial for fully grasping the concept. The use of symbols, like open and closed circles on the number line, tell us whether endpoint values are included.
Interval notation is a concise way to express set solutions:

• Parentheses \((...)\) are used when a number is not included, which we refer to as an open interval. For example, in \(-2 < x\), \(-2\) uses a parenthesis: \((-2\).
• Brackets \([...]\) show the inclusion of the number, known as a closed interval. Thus, \(x \leq 1\) is written as \([1]\).
• Combining them for our exercise, the interval \((-2, 1]\) means numbers greater than \(-2\) but up to and including \(1\).

Graphing and interval notations complement each other. Learning both is helpful for solving mathematical problems involving ranges and constraints efficiently.