Problem 39 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x>-2 \text { and } x<3 $$

Step-by-Step Solution

Answer

The interval is \((-2, 3)\).

Step 1: Understand the Inequalities

In this exercise, we are given two inequalities: \( x > -2 \) and \( x < 3 \). We are tasked with finding the values of \( x \) that satisfy both conditions simultaneously.

Step 2: Identify the Intersection of Conditions

Since the conditions must both be true, we need to find the range of \( x \) that satisfies both inequalities. This is the intersection of \( x > -2 \) and \( x < 3 \). Therefore, the combined solution is \(-2 < x < 3\).

Step 3: Graph the Solution on a Number Line

Draw a horizontal number line. Mark the point \(-2\) and draw an open circle to indicate that \( x \) is greater than \(-2\) but does not include \(-2\). Similarly, mark the point \(3\) and draw an open circle to indicate that \( x \) is less than \(3\) but does not include \(3\). Shade the region between \(-2\) and \(3\) to show the solutions.

Step 4: Write the Interval Notation

The solutions to the inequalities \( x > -2 \) and \( x < 3 \) can be expressed in interval notation as \((-2, 3)\). This notation uses parentheses to show that the endpoints \(-2\) and \(3\) are not included in the solution set.

Key Concepts

number line

A number line is a simple graphical representation of numbers in a straight, horizontal line format. It helps us easily visualize the relationships between numbers, especially when dealing with inequalities. Markings are usually made at equal intervals, representing values, while specific points on the line correspond to real numbers.

In our example, for the inequality conditions given by \( x > -2 \) and \( x < 3 \), the number line is useful to depict the range of acceptable values.

To represent \( -2 < x < 3 \), you need to:

• Draw a horizontal line and label the numbers -2 and 3 on it.
• Place open circles or dots (not shaded) at -2 and 3. This indicates that -2 and 3 themselves are not included in the solution.
• Shade the line between these two points to show all numbers greater than -2 and less than 3.

The use of open circles and shading visually indicates which numbers satisfy the inequality.

interval notation

Interval notation is a mathematical way to describe subsets of real numbers, focusing on the values that a variable could take, especially when dealing with continuous intervals on a number line.

In our exercise example, we need to express \( -2 < x < 3 \) in interval notation.

Interval notation uses parentheses and brackets to indicate whether endpoints are included or excluded from the solution set:

• Parentheses \(( )\) are used when the endpoint is not included.
• Brackets \([ ]\) are used when the endpoint is included.

Thus, the solution \( -2 < x < 3 \), which excludes -2 and 3, can be written as \((-2, 3)\). The parenthesis signifies that neither endpoint is part of the solution set.

solution set

A solution set is a collection of all possible values that satisfy a given inequality or equation. It represents where all conditions imposed by the inequalities are true.

In the given problem, you have two conditions: \( x > -2 \) and \( x < 3 \). The solution set is the values of \( x \) that make both inequalities true simultaneously.

Here's how you determine it:

• Identify each condition separately. The first tells you \( x \) must be greater than -2. The second indicates \( x \) must be less than 3.
• Intersect these conditions to find the range covering both, which is \( -2 < x < 3 \).

The solution set is then represented both graphically on the number line and algebraically in interval notation, clarifying the range of possible values \( x \) can take that satisfy both conditions. This clear representation helps anyone reading the solution understand which values are included.