Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Express the inequality as an interval, and sketch its graph. $$ -3 \leq x<5 $$

Step-by-Step Solution

Answer

The interval is \([-3, 5)\), including \(-3\) and not including 5.

Step 1: Identify the Inequality Expression

The given inequality is \[-3 \leq x < 5\]This indicates that \(x\) is greater than or equal to \(-3\) and less than 5.

Step 2: Express the Inequality as an Interval

The interval notation for the inequality \(-3 \leq x < 5\) is \([-3, 5)\). This means the interval includes \(-3\) (closed bracket) and goes up to, but does not include, 5 (open bracket).

Step 3: Sketch the Graph

To graph the interval \([-3, 5)\):- Draw a number line.- Mark the point \(-3\) with a filled-in circle to indicate that it is included in the interval.- Mark the point 5 with an open circle to indicate that it is not included.- Shade the region on the number line from \(-3\) to just before 5. This shaded region represents all the values that \(x\) can take within the interval.

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers on the real number line. It is used particularly for expressing inequalities in a concise format.
When using interval notation, it is important to distinguish between different boundary types:

• A bracket "[" or "]" means that the number is included in the interval, also known as a closed interval.
• A parenthesis "(" or ")" indicates the number is not included in the interval, creating an open interval.

Consider \(-3 \leq x < 5\), which in interval notation is written as \([-3, 5)\). In this case:

• \(-3\) is included in the interval, hence the close bracket "[".
• \(5\) is not included, so a parenthesis symbol is used ")".

Using interval notation helps in clearly communicating which numbers are part of a solution set, removing ambiguity that often accompanies wordy descriptions of integers.

Number Line Graphing

Graphing inequalities on a number line provides a visual means of representing solution sets. This approach allows you to quickly understand which numbers are included in a particular interval.
To graph an interval like \([-3, 5)\), follow these steps:

• Draw a long horizontal line to represent all real numbers.
• Mark the points \(-3\) and \(5\) on this line.
• Place a filled-in circle on \(-3\) since \(-3\) is included in the interval, indicating a closed boundary.
• Place an open circle on \(5\) to show that this number is not part of the interval.
• Shade the region between \(-3\) and \(5\) to clearly illustrate all the values that fall within the interval.

This shaded area is important because it represents all the possible values of \(x\). By presenting inequalities through number line graphing, you simplify complex scenarios into an easily understandable visual format.

Closed and Open Intervals

Closed and open intervals determine whether the boundaries of an interval are included or excluded from the solution set.
Understanding the difference can fundamentally alter the interpretation of inequalities.

• Closed Intervals: Denoted by square brackets "[a, b]". This means both endpoints, \(a\) and \(b\), are included. For example, \([-3, 5]\) includes all values from \(-3\) to \(5\), including both endpoints.
• Open Intervals: Signified by parentheses "(a, b)". Neither endpoint \(a\) nor \(b\) is included in the interval. For instance, \((3, 7)\) includes values greater than \(3\) and less than \(7\), but not the numbers \(3\) and \(7\) themselves.
• In the given exercise, \([-3, 5)\) represents an interval that includes \(-3\) (closed interval at the left) but does not include \(5\) (open interval at the right).

These intervals provide a precise and streamlined way to describe number sets, enhancing clarity in mathematical expressions.