Problem 3 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ 4 \leq x<19 $$

Step-by-Step Solution

Answer

[4, 19)

Step 1: Identify the Inequality Type

We have the compound inequality: \(4 \leq x < 19\). It combines two inequalities: \(4 \leq x\) and \(x < 19\). This means \(x\) is bounded below by 4 (inclusive) and above by 19 (exclusive).

Step 2: Determine Boundaries and Bracket Types

In interval notation, the lower boundary '4' is included because of the 'less than or equal to' sign, so we use a bracket '[' to denote inclusion. The upper boundary '19' is not included because of the 'less than' sign, so we use a parenthesis ')' to denote exclusion.

Step 3: Write the Interval Notation

Given the conditions determined in the previous step, the interval notation representing \(4 \leq x < 19\) is written as \([4, 19)\). This captures all the numbers \(x\) that are greater than or equal to 4 and strictly less than 19.

Key Concepts

Compound Inequality

In mathematics, a compound inequality involves two separate inequalities combined into one statement. This helps to define a range of values that a variable can take. In our example, the compound inequality is \(4 \leq x < 19\). This means that \(x\) must satisfy both conditions in order to be true.

• The first part, \(4 \leq x\), tells us that \(x\) has to be at least 4, and any value larger than 4 will also work.
• The second part, \(x < 19\), indicates that \(x\) has to be less than 19.

These conditions are joined by the word "and." So, both need to be true for values of \(x\). This way, compound inequalities allow us to express conditions where \(x\) lies within a specific range.

Inequality Type

Understanding inequality types is crucial in knowing how to state conditions accurately. The inequality \(4 \leq x < 19\) is a mixed inequality, as it combines two different types of inequalities.
When we see a \(\leq\) sign, it means "less than or equal to," signifying inclusion. This suggests that the number that is associated with the inequality is included within the solution set.
With a "<" sign, it means strictly "less than," indicating that the upper boundary isn't included in the solution. Recognizing these types helps in determining how the range of values will be presented in interval notation.

Boundaries and Bracket Types

When writing interval notation, it's important to reflect the boundaries of the variable and use the correct bracket type. For the inequality \(4 \leq x < 19\), we have:

• The lower boundary is 4. Since the inequality \(4 \leq x\) includes 4, we use a square bracket "[" to indicate inclusion.
• The upper boundary is 19. However, because \(x < 19\), 19 is not included in the range. Consequently, we use a parenthesis ")" to show exclusion.

So, the interval notation correctly reflecting the inequality \(4 \leq x < 19\) is \([4, 19)\). Understanding the symbols and bracket types helps make these conversions between inequalities and interval notation straightforward.

Algebra

Interval notation is a part of algebra, used to succinctly express the range of solutions for inequalities. Algebra helps to solve and understand equations and inequalities, which are the building blocks of mathematical analysis.
When dealing with algebraic expressions such as inequalities, representing them in interval notation provides a clean, concise way to communicate complex ideas. It helps to visually interpret the scope of variables and how they interact within constraints.
In practice, this knowledge makes it easier to solve equations and understand where solutions lie. It transforms the abstract symbols of algebra into tangible ranges that are easier to grasp, aiding in problem-solving and mathematical reasoning.