Problem 19 - Algebra for College Students

Question

For Problems 19-34, graph the solution set for each compound inequality, and express the solution sets in interval notation. \(x>-1 \quad\) and \(\quad x<2\)

Step-by-Step Solution

Answer

The solution in interval notation is \((-1, 2)\).

Step 1: Understand the Inequality

The compound inequality given is \(x > -1\) and \(x < 2\). This means we need to find values of \(x\) that satisfy both conditions at the same time.

Step 2: Plot the Inequalities on a Number Line

First, we plot \(x > -1\) on a number line, which means that we place an open circle at -1 and shade all numbers to the right. Next, we plot \(x < 2\) on the same number line, which means placing an open circle at 2 and shading all numbers to the left. The overlapping region represents the solution of the compound inequality.

Step 3: Identify the Overlapping Region

The overlapping region on the number line is between -1 and 2, but not including -1 and 2 themselves, since both circles are open.

Step 4: Write the Solution in Interval Notation

The solution set for the compound inequality \(x > -1\) and \(x < 2\) is represented in interval notation as \((-1, 2)\). This notation signifies all numbers between -1 and 2, not including -1 and 2 themselves.

Key Concepts

Inequality Expressions

Compound inequalities combine two or more simple inequalities into a single statement by using the words "and" or "or." In this exercise, we see the compound inequality \(x > -1\) and \(x < 2\). The word "and" signifies that both conditions must be true simultaneously for a number to be part of the solution set.

In inequality expressions, symbols like \(>\), \(<\), \(\geq\), and \(\leq\) are used to represent relationships between values.

• \(x > -1\) indicates that any value of \(x\) must be greater than \(-1\) but not equal to it.
• \(x < 2\) signifies that \(x\) must be less than \(2\) but not equal to it.

Thus, for \(x\) to satisfy the compound inequality \(x > -1\) and \(x < 2\), it must be greater than \(-1\) while simultaneously being less than \(2\). This clearly highlights the need to fulfill both parts of the compound statement.

Interval Notation

Interval notation provides a concise way to express the range of values an inequality covers. For the compound inequality \(x > -1\) and \(x < 2\), we translate these conditions into interval notation to simplify communication.Interval notation uses parentheses \(()\) and brackets \([]\) to indicate whether endpoints are included. Parentheses mean the endpoint is not included, while brackets mean it is included.

Here, since \(-1\) and \(2\) are not included as solutions (indicated by \(<\) and \(>\) in the inequalities), we use parentheses:

• \((-1, 2)\) represents all numbers between \(-1\) and \(2\), excluding the endpoints themselves.

This format tells us that any number inside \(-1\) and \(2\) is a solution, simplifying the representation and making it easier to handle mathematically.

Number Line Representation

Representing inequalities on a number line is a helpful visual method to see solution sets. It provides a clear illustration of what values are included within an expression.When we graph \(x > -1\) and \(x < 2\) on a number line, we begin by focusing on each inequality separately:

• To graph \(x > -1\), we draw an open circle at \(-1\) and shade all values to the right.
• For \(x < 2\), we place an open circle at \(2\) and shade all values to the left.

The solution set for the compound inequality is the overlapping shaded region between \(-1\) and \(2\), excluding \(-1\) and \(2\) themselves because of the open circles. This visually demonstrates which values satisfy both conditions and is a useful way to confirm the range of solutions you have determined algebraically.