Problem 44 - Algebra and Trigonometry

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$ -5 < x < 2 $$

Step-by-Step Solution

Answer

The inequality \(-5 < x < 2\) in interval notation is \((-5, 2)\), and it is represented by an open interval on the number line from \(-5\) to \(2\).

Step 1: Understand the Inequality

The inequality given is \(-5 < x < 2\). This means that \(x\) is a number greater than \(-5\) and less than \(2\).

Step 2: Translate Inequality to Interval Notation

In interval notation, an inequality of the form \(-5 < x < 2\) is expressed as \((-5, 2)\). The round brackets indicate that \(-5\) and \(2\) are not included in the interval.

Step 3: Graph the Interval

On a number line, draw a line between \(-5\) and \(2\). Use open circles at \(-5\) and \(2\) to indicate these values are not included in the interval. Shade the region between these open circles to represent all numbers between \(-5\) and \(2\).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In our example, \(-5 < x < 2\), this tells us that \(x\) is somewhere between \(-5\) and \(2\). It is not equal to these values but larger than \(-5\) and smaller than \(2\).
Unlike equations, which have one specific solution, inequalities represent a range of possible values.
Understanding how to work with inequalities is crucial as they appear frequently in algebra, calculus, and many real-life applications.

• "Less than" is represented by \(<\) and "more than" by \(>\).
• "Less than or equal to" combines the symbol \(<\) with an equal sign, represented as \(\leq\).
• Similarly, "greater than or equal to" is shown by \(\geq\).

These operators are important for defining ranges and boundaries in mathematical solutions.

Using the Number Line

The number line is a visual way to describe and work with numbers. It's particularly helpful when graphing inequalities, as it shows where numbers fall in relation to each other.
To graph \(-5 < x < 2\) on a number line, we start by placing marks at \(-5\) and \(2\). These marks help us understand where the range begins and ends.
This range includes all numbers between these two marks but not the numbers themselves.

• Mark the number line with clear increments, making sure critical points like \(-5\) and \(2\) are distinct.
• Ensure that the part of the number line between these two points is clear and can be shaded or highlighted.

Visualizing an inequality on a number line is a straightforward way to grasp which numbers are solutions. It simplifies understanding by turning an abstract concept into something you can see.

Significance of Open Circles

Open circles are used on number lines to indicate values that are not included in an interval. In the inequality \(-5 < x < 2\), both \(-5\) and \(2\) are not part of the solution set.
Open circles at these positions on the number line perfectly show this exclusion.
In interval notation, we also use parentheses \((-\) and \()\)) to denote this fact.

• The open circle at \(-5\) tells us that while values just above \(-5\) are included, \(-5\) itself is not.
• Similarly, the open circle at \(2\) marks that all values less than but not equal to \(2\) are part of the solution.

This use of open circles is key to accurately showing which numbers are considered part of a solution and which are not, ensuring the graphically depicted range is precise and clear.