Problem 8 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ 17>x \geq-3 $$

Step-by-Step Solution

Answer

Interval notation: \([-3, 17)\).

Step 1: Understand the Inequality

The given inequality is \(17 > x \geq -3\). This inequality means that \(x\) is less than 17 and simultaneously greater than or equal to \(-3\).

Step 2: Identify the Boundary Points

The boundary points for \(x\) based on the inequality are 17 and \(-3\). The inequality includes \(-3\) but does not include 17, as indicated by the symbols \(\geq\) and \(>\).

Step 3: Create Interval Notation

Since \(x\) can include \(-3\) but not 17, the correct interval notation uses a square bracket at \(-3\) and a parenthesis at 17. Therefore, the interval is written as \([-3, 17)\).

Key Concepts

inequalities

In mathematics, inequalities are statements about the relative size or order of two objects, typically involving variables. They show how one quantity relates to another, which could be greater than, less than, greater than or equal to, or less than or equal to. In the example inequality given (17 > x \geq -3), we're looking at a range of values for \(x\) that lie between two specified numbers.
The inequality symbols tell us the nature of these boundaries:

• \(>\) means greater than, so the value is less than 17.
• \(\geq\) means greater than or equal to, so the value is at least -3.

Understanding how each part of the inequality impacts the range is crucial for correctly interpreting it in interval notation. It highlights which parts of the boundary are inclusive, as well as the overall direction of the inequality, which in this case runs from -3 up to, but not including, 17.

boundary points

Boundary points are specific values in an inequality that define where the range of possible solutions lies. They mark the borders of intervals, indicating which values \(x\) can take. In the inequality \(17 > x \geq -3\), the boundary points are 17 and -3.

These points inform us where the solution set starts and ends:

• -3 is marked as an inclusive point because of the \(\geq\) symbol. This means -3 is part of the solution set. So, it will receive a bracket [ in interval notation.
• 17 is not included in the solution set because it is marked by the \(>\) sign. Thus, it uses a parenthesis ) to show exclusion.

Understanding boundary points helps in writing the correct interval notation, as you need to know if these boundary values themselves are considered solutions.

algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's about finding unknown values using known values and operations, which often involve solving equations or inequalities.
In dealing with inequalities like \(17 > x \geq -3\), algebra provides a structured way to express relationships between variables and constants. The solution is to use operations that preserve the direction of the inequality while finding valid ranges for the variable \(x\).

• Applying operations like addition, subtraction, multiplication, or division to both sides of the inequality.
• Carefully handling inequalities, as multiplying or dividing by a negative number reverses the inequality.

Knowing algebraic manipulation is vital for solving and expressing inequalities beyond just interval notation; it helps in proving and understanding mathematical relationships.