Problem 16 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$[-4,3)$$

Step-by-Step Solution

Answer

The interval [-4,3) can be expressed as the inequality -4 ≤ x < 3. The graph of this interval on a number line starts from -4 (inclusive, represented by a filled circle) to 3 (exclusive, represented by an open circle).

Step 1: Interpret the Interval

The interval given is [-4,3). In interval notation, square brackets mean the boundary is included, and round brackets mean it is not. Therefore, this interval represents all numbers between -4 and 3, including -4 but not including 3.

Step 2: Convert Interval to Inequality

The interval [-4,3) can be expressed as an inequality as such: -4 ≤ x < 3. Here, 'x' denotes the variables that are part of this interval. This inequality simply means that 'x' can take any value that is greater than or equal to -4 but less than 3.

Step 3: Draw the Number Line

First, draw a horizontal line and mark a simple scale on it. The scale should cover all the numbers in the inequality.

Step 4: Plot the Interval

Mark a solid dot on -4 because -4 is included in the interval, and then draw a line from -4 to 3. Use an open circle to indicate 3 because 3 is not included in the interval.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is larger or smaller than the other. Instead of using the equals sign, inequalities use symbols like:

• < : less than
• ≤ : less than or equal to
• > : greater than
• ≥ : greater than or equal to

Inequalities can be read just like sentences, from left to right. For example, "5 < x" means "5 is less than x," or equivalently, "x is greater than 5." This concept is crucial when expressing intervals, as it translates numerical ranges into simple mathematical terms that describe all possible solutions.

Plotting on a Number Line

A number line is a visual tool used for representing numbers on a straight line, where each point corresponds to a number. Visualizing inequalities on a number line helps you quickly understand which numbers satisfy the inequality. Here's how you can plot an interval like \([-4, 3)\) on a number line:

• First, draw a horizontal line. This becomes your number line.
• Mark important points from the interval, in this case, -4 and 3.
• Use a solid dot to show that -4 is included in the interval.
• Draw an open circle at 3 to indicate it is not included, completing the visual representation.

This number line representation assists in comprehending which values are eligible based on the given inequality.

Defining a Closed Interval

Closed intervals are a way to express a range of numbers where both endpoints are included in the set. In interval notation, a closed interval is denoted with square brackets. For example, \([-4, 3]\) would mean that the interval includes every number between -4 and 3 as well as -4 and 3 themselves.

• The notation tells us to include the boundaries
• This is different from open intervals, where boundaries are excluded

You can think of closed intervals as a closed door: everything inside, including what’s touching the door, is part of the space.

Understanding an Open Interval

An open interval refers to a range of numbers that excludes its endpoints. In interval notation, this is marked with parentheses. A key detail to remember is that although the interval covers the values between the numbers, the precise endpoints themselves are not part of the interval.

• For example, in the interval \(3, 7\), the numbers 3 and 7 are not included.
• This tells us the range includes 3.0001 or 6.9999, just not 3 or 7 themselves.

This can be imagined like an open fence, where everything inside the bounds can pass but not the endpoints.