Problem 13 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ -2

Step-by-Step Solution

Answer

The solutions are graphed between open circles at -2 and 5, with interval notation \((-2, 5)\).

Step 1: Understanding the Inequality

This inequality tells us that the variable \( x \) must be between -2 and 5. It does not include -2 or 5 as part of the solution, as indicated by the strict inequalities \(<\).

Step 2: Drawing the Number Line

Draw a horizontal line and mark the significant values from the inequality: -2 and 5. Make sure these points are accurately placed relative to other numbers on the line.

Step 3: Marking the Points

On the number line, place an open circle at both -2 and 5. The open circle indicates that these points are not included in the solution set.

Step 4: Shading the Solution Region

Shade the portion of the number line between the open circles at -2 and 5. This shaded region represents all the numbers that are solutions to the inequality \(-2 < x < 5\).

Step 5: Writing the Interval Notation

In interval notation, use parentheses to denote that -2 and 5 are not included. Write \((-2, 5)\) as the interval notation for the solutions.

Key Concepts

Number Line

Visualizing inequalities becomes easier with a number line. It's a simple graphical representation that helps you see where all solutions to an inequality lie. To begin, draw a horizontal line and place relevant numbers on it. For our problem, crucial values are -2 and 5. This scale should be consistent so that it's clear where other numbers fall relative to these points.

Using a number line, you can easily mark specific points and regions that represent solutions. It's a great way to visualize whether particular numbers are included or excluded from the solution set. This helps in understanding inequalities better and is an essential step before moving to interval notation.

Interval Notation

Interval notation is a concise way to describe a set of numbers. Here, we deal with strict inequalities where -2 and 5 are not included. In such cases, we use parentheses, not brackets. The interval notation for our inequality (-2 < x < 5) is \((-2, 5)\).

Here's how interval notation works:

• Parentheses \((\text{ or } )\) are used when endpoints are not included.
• Brackets \([\text{ or } ]\) would be used if endpoints were included.

With interval notation, you can write complex inequality solutions efficiently. It's quick, clear, and widely accepted in math.

Practicing this notation helps you understand how numbers fit together in a solution set.

Open Circle

An open circle on a number line signifies that a particular number is not included in the solution set. This is a crucial part of graphing inequalities. For our inequality \(-2 < x < 5\), open circles are placed at -2 and 5.

The open circle tells you:

• That the point is a boundary, but not part of the solution.
• Which specific numbers should not be considered as part of the set.

Without open circles, it might seem like -2 and 5 are included, which would be incorrect. This distinction using open circles prevents misunderstanding, ensuring you accurately portray the inequality's solution on the number line.