Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Express the inequality as an interval, and sketch its graph. $$ -3

Step-by-Step Solution

Answer

The interval is \((-3, -1)\).

Step 1: Understand the Inequality

The given inequality is \(-3 < x < -1\). This means that \(x\) is a value that is greater than \(-3\) but less than \(-1\). Both \(-3\) and \(-1\) are not included in the set of solutions, meaning they are open endpoints.

Step 2: Write the Inequality as an Interval

Intervals show the range of values that \(x\) can take. Since \(x\) is greater than \(-3\) and less than \(-1\), we use round brackets to indicate that the endpoints are not included. The interval notation for \(-3 < x < -1\) is \((-3, -1)\).

Step 3: Sketch the Graph of the Interval

To sketch the graph of \((-3, -1)\), draw a number line. Mark the points \(-3\) and \(-1\) with open circles to indicate they are not included in the solution. Shade the region between \(-3\) and \(-1\) to represent all the values \(x\) can take within this interval.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions involving a range of values. Simply put, they tell us how one variable relates to another using symbols like `<`, `>`, `≤`, or `≥`. The inequality \(-3 < x < -1\) states that the variable \(x\) must be greater than \(-3\) and less than \(-1\). In other words, \(x\) can be any number between \(-3\) and \(-1\), but not equal to these endpoints.

• Less than (\(<\)) or greater than (\(>\)) without an equal sign indicates the number is not included.
• Less than or equal to (\(\leq\)) or greater than or equal to (\(\geq\)) means the number is included in the solutions.

Remember, in inequalities like these, the endpoints don't form part of the solution set.

Using the Number Line

A number line is a great visual tool for understanding inequalities. It is a straight line with numbers placed at intervals along its length. It helps us easily identify and represent the range of possible values for variables like \(x\). In this exercise, we use the number line to graphically show all values for \(x\) in the range from \(-3\) to \(-1\). The numbers are laid out in increasing order. Where \(-3\) is to the left, followed by numbers up to just before \(-1\). This visual representation provides a quick and easy way to see the solution. You can think of it as zooming in on a section of the number line where all possible values sit snugly between specific points.

Open Endpoints Explained

Open endpoints in inequality signify that certain boundary values are not part of the solution set. When intervals are written using parentheses, as in \((-3, -1)\), it shows open endpoints. Here, \(-3\) and \(-1\) are the endpoints that \(x\) cannot exactly reach.

• The parenthesis (\((\) or \()\)) tells us that the endpoint is not included.
• Brackets (\([\) or \()]\)) would mean the endpoint is included.

Visually, in graph form, these are represented with open circles on the number line. This signifies that while close, the boundary points themselves are not parts of the solutions.

Graphing Intervals on the Number Line

Graphing intervals on a number line involves illustrating the solution of an inequality. Using \(-3 < x < -1\), we begin by marking appropriate points \(-3\) and \(-1\) on the number line.To denote that these points are not included, open circles are placed on \(-3\) and \(-1\). The space in between these points is then shaded or highlighted to indicate all valid values of \(x\) that fall within but do not include the endpoints. It’s like coloring in all the valid \(x\) options, defined by your inequality.

• Mark open circles to exclude values.
• Shade the line between points to include those numbers in the set.

Graphing this way gives a clear and precise picture of the solutions within a range, making it easy to understand at a glance.