Problem 4 - Algebra and Trigonometry

Question

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. $$ [-4,3) $$

Step-by-Step Solution

Answer

The interval \([-4,3)\) is represented in set-builder notation as \(x: -4 \leq x < 3\). When graphed on a number line, we draw a closed dot at -4 and an open dot at 3, with the line in between representing the numbers in this interval.

Step 1: Write the interval in set-builder notation

In set-builder notation, the interval can be denoted as: \(x: -4 \leq x < 3\). This notation reads as 'the set of all x such that x is greater than or equal to -4 and less than 3'.

Step 2: Graph the interval on a number line

To graph this on a number line, you start by drawing a line. Next, mark off the points at -4 and 3. A closed (filled in) dot is placed on -4 to indicate that -4 is included in the interval. An open (not filled in) dot is placed on 3 to indicate that 3 is not included in the interval, and an arrow is drawn to the right of 3 pointing left to show all numbers less than 3 are included. The section of the line between -4 and 3 represents the interval.

Key Concepts

Interval Notation

Understanding interval notation is vital for conveying ranges of values succinctly in mathematics. Interval notation describes a set of numbers lying between a defined lower and upper bound. For example, the interval \( [-4,3) \) includes all real numbers from -4 to slightly less than 3. Here's how to interpret it:

The square bracket \( [ \) indicates that the lower bound, -4, is included in the set. On the other hand, the parenthesis \( ) \) signifies that the upper bound, 3, is excluded. So although -4 is part of the set, 3 is not, and the set encompasses all real numbers in between these boundaries.

To improve understanding of interval notation, remember:

• Inclusive boundaries are denoted by square brackets \( [ \) or \( ] \).
• Exclusive boundaries are denoted by parentheses \( ( \) or \( ) \).
• Intervals can involve infinity, indicated with a parenthesis, such as \( (a, +\infty) \) or \( (-\infty, b] \) where numbers towards infinity are naturally never included.

These basics make written communication of ranges in mathematics clear and efficient.

Number Line Graphing

Visualizing intervals can become more intuitive with number line graphing. Number lines are a foundational tool in mathematics that allow us to graphically represent intervals and sets of real numbers. Referring to the interval \( [-4,3) \) from the earlier problem, drawing a number line can aid in clearly seeing which numbers are part of the set and which aren't.

To graph, draw a horizontal line representing all possible real numbers. Mark the interval's boundaries, -4 and 3, on the line: a solid dot on -4 because it's included (remember the square bracket?), and an open dot on 3 since it's not part of the interval (recall the parenthesis?). Connect the dots to highlight the set, and voilà, we have a visual representation of the interval \( [-4,3) \).

Remember these graphing tips:

• Use solid dots for inclusive boundaries and open dots for exclusive boundaries.
• A number line should extend beyond the values of your interval to provide context.
• Arrows can indicate the set extends indefinitely if infinity is involved in the interval.

Inequalities in Mathematics

The concept of inequalities is a fundamental part of understanding how intervals operate within mathematics. Inequalities help us describe mathematical relationships where quantities are not equal but rather greater or less than one another.

In set-builder notation, we use inequalities to define a set of numbers in a concise way. For instance, the expression \(x: -4 \leq x < 3\) translates the interval \( [-4,3) \) into a statement that includes all x values that are greater than or equal to -4 but less than 3.

Here are a few tips for grasping inequalities:

• \(<\) and \(>\) mean 'less than' and 'greater than' and do not include the number itself.
• \(\leq\) and \(\geq\) stand for 'less than or equal to' and 'greater than or equal to', indicating the number is part of the set.
• Always read inequalities from the variable, such as in \(x < 3\), where x is any number less than 3.
• Remember that inequalities are used to build ranges on the number line, which ties back to the graphical representation.

Inequalities make the analysis of number sets and mathematical reasoning not only possible but clearer, especially when solving problems involving ranges of values.