Problem 56 - Precalculus Mathematics for Calculus

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$-5

Step-by-Step Solution

Answer

The interval is \((-5, 2)\) with the region between open circles at \(-5\) and \(2\) shaded.

Step 1: Understand the Inequality

The inequality given is \(-5 < x < 2\). This means \(x\) is greater than \(-5\) but less than \(2\). There are no equality signs, so \(x\) cannot equal \(-5\) or \(2\).

Step 2: Express in Interval Notation

Interval notation is a way of writing the set of all numbers between two endpoints. Since \(x\) is greater than \(-5\) and less than \(2\), but not equal to either, we use parentheses. In interval notation, this is written as \((-5, 2)\).

Step 3: Graph the Interval on a Number Line

To graph \((-5, 2)\), draw a number line and place open circles at \(-5\) and \(2\) since the endpoints are not included. Shade the region between \(-5\) and \(2\) to indicate all numbers between these values are part of the solution.

Key Concepts

Understanding Inequalities

Inequalities describe a range of possible values that a variable can take. They use symbols like <, >, ≤, and ≥ to show relationships between variables or numbers. In the inequality \(-5 < x < 2\), we know that the variable \( x \) is greater than \(-5\) and less than \( 2 \). However, since there are no "equals" components (i.e., no \(≤ \) or \(≥ \)), \( x \) cannot include the end values \(-5\) and \( 2 \).

Inequalities are useful because they show ranges of solutions rather than exact numbers. They're foundational in algebra, helping describe conditions where something must be larger, smaller, or simply different by some quantity.

Graphing with a Number Line

A number line graph is a simple way to represent inequalities visually. It helps to see where a set of numbers fall in relation to each other. For the inequality \(-5 < x < 2\), a number line provides clarity on the range of values for \( x \.\)

To graph this on a number line:

• Draw a straight, horizontal line and mark the numbers of interest, in this case, \(-5\) and \( 2 \.\)
• Place an open circle, or an empty dot, at each endpoint: on \(-5\) and \( 2\). The open circle indicates that these values are not included, which directly relates to the inequality not having equal components.
• Shade or color the section of the number line between these open circles. This visually represents all numbers between \(-5\) and \( 2\), excluding the endpoints.

This method reinforces understanding of which values solve the inequality and makes it easier to communicate these solutions visually.

Introduction to Open Intervals

An open interval is a range of numbers between two endpoints where neither endpoint is included in the interval. It is represented using parentheses, like \((-5, 2)\), which corresponds to our inequality. Parentheses in interval notation indicate openness and tell you that the boundary numbers themselves are not part of the solution.

Open intervals are particularly helpful since they simplify the notation of "greater than" and "less than" inequalities. They are a concise way to express the "between" concept in math.

Here's a quick summary to keep it in mind:

• Open intervals use round brackets: ( ) instead of square brackets [ ], which define closed intervals where endpoints are included.
• They clearly exclude the boundary points, unlike closed intervals that might have equal signs in the inequalities.
• Such notation saves time and offers a quick visual cue as to which numbers are part of the solution set.

Understanding open intervals helps one navigate and solve many types of mathematical problems more efficiently.