Problem 17 - College Algebra

Question

Express interval in set-builder notation and graph the interval on a number line. \((-3,0) \cup[-1,2]\)

Step-by-Step Solution

Answer

The set-builder notation for the given interval is \{x: -3 < x < 0\} \(\cup\) \{x: -1 \leq x \leq 2\}. The graph of this interval on a number line will have an open circle at -3 and 0, representing the first interval, and a closed circle with a shaded line in-between at -1 and 2, representing the second interval.

Step 1: Identify the given intervals

Two intervals are given here: \((-3,0)\) and \([-1,2]\). The parenthesis () in interval notation means the end points are not included in the interval while the brackets [] mean that the end points are included.

Step 2: Convert to Set-Builder Notation

We convert the given intervals into set-builder notation. First interval becomes \{x: -3 < x < 0\}. Second interval becomes \{x: -1 \leq x \leq 2\}.

Step 3: Union of Sets

Here, the union \(\cup\) of the two sets is taken, which combines the two sets into one. Any element that is part of either or both of the original sets will be in the union. So, the union set becomes \{x: -3 < x < 0\} \(\cup\) \{x: -1 \leq x \leq 2\}.

Step 4: Draw the number line

To illustrate the given intervals, draw a number line. For the interval \((-3,0)\), plot an open circle at -3 and 0 and shade the line between these two points. This indicates all numbers between -3 and 0 are part of this interval. For the interval \([-1,2]\), plot a closed circle at -1 and 2 and shade the line between these two points. This includes the numbers between -1 and 2, including the end points -1 and 2 themselves.

Key Concepts

Interval Notation

Interval notation is a way to describe sets of numbers between given endpoints. It uses parentheses or brackets to indicate whether the endpoints are included in the interval.

• Parentheses \( ( \) indicate that an endpoint is not included. For example, \((-3, 0)\) represents all numbers greater than -3 and less than 0, but not -3 or 0 themselves.
• Brackets \( [ \) mean that the endpoint is included. So \[ -1, 2 \] includes the numbers -1 and 2, as well as everything in between.

Interval notation is a concise way to specify a range of values, making it especially useful in mathematical expressions and functions. Understanding how to interpret the symbols is key to solving problems related to ranges and sets.

Union of Sets

The union of sets is a concept in mathematics that combines all elements from multiple sets. The symbol for union is \( \cup \). It is used to join sets together, forming a new set containing all unique elements.

• For the intervals \((-3,0)\) and \[ -1, 2 \], the union is \{x: -3 < x < 0\} \cup \{x: -1 \leq x \leq 2\}\. This means any number that belongs to either the first set or the second set is included.
• The union operation does not repeat elements already present in both sets. It merely considers them once in the resulting set.

Using union is helpful when dealing with compound inequalities or when you need to combine solutions from different conditions.

Number Line

A number line is a visual representation of numbers on a straight line, which helps to understand and solve problems related to intervals and inequalities. It's a crucial tool for illustrating concepts visually.

• To graph \((-3,0)\), draw an open circle at -3 and 0 to show they are not included, then shade the line between them.
• For \[ -1, 2 \], use closed circles at -1 and 2. Shade the line segment between these points to include the endpoints and all numbers in between.

A number line not only makes abstract interval concepts tangible but also helps students visualize solutions for inequalities and equations, reinforcing understanding through seeing and doing.