Problem 48 - Algebra and Trigonometry

Question

\(47-52\) : Express the inequality in interval notation, and then graph the corresponding interval. $$ 1 \leq x \leq 2 $$

Step-by-Step Solution

Answer

Interval notation is \([1, 2]\); graph has solid circles at 1 and 2.

Step 1: Understand the Inequality

The inequality given is \( 1 \leq x \leq 2 \). This means that \( x \) is greater than or equal to 1 and less than or equal to 2. Both endpoints 1 and 2 are included in this interval.

Step 2: Express in Interval Notation

Since both endpoints are included in the inequality, we use square brackets to denote the interval. The interval notation for \( 1 \leq x \leq 2 \) is \([1, 2]\).

Step 3: Graph the Interval

To graph the interval \([1, 2]\), draw a number line. Mark the points at 1 and 2 and draw a solid line connecting these points. Place solid circles at both 1 and 2 to indicate that these points are included in the interval.

Key Concepts

Interval Notation

Interval notation is a concise way of representing a set of numbers, typically numbers on a number line. It is especially useful when dealing with inequalities. When expressing an inequality such as \( 1 \leq x \leq 2 \), two key symbols are used: parentheses \(( )\) and square brackets \([ ]\).

• Square brackets are used when the endpoints are included in the interval, indicating closed intervals. For example, \([1, 2]\) means both 1 and 2 are included.
• Parentheses are used for open intervals when endpoints are not included. For example, \((1, 2)\) means 1 and 2 are not included.

In the given inequality \(1 \leq x \leq 2\), both endpoints 1 and 2 are part of the solution set, so we use square brackets: \([1, 2]\) is the interval notation. This concise method helps simplify the representation of solutions to inequalities.

Number Line Graphing

Graphing on a number line provides a visual representation of an interval, complementing interval notation. To graph the interval \([1, 2]\), follow these simple steps:

• Draw a horizontal line—this represents the number line, with small vertical marks representing numbers.
• Identify and mark the endpoints of the interval. In this exercise, the points 1 and 2 are marked on the line as they are the boundaries of the interval.
• Connect these points with a solid line to show all the points between 1 and 2 are included in the interval.
• Add solid circles or dots on the endpoints to signify that these values are part of the interval. This indicates a closed interval, matching the square brackets used in interval notation.

The number line graph is a quick visual reference that represents all possible values \(x\) can take, spanning from 1 to 2, inclusive of both endpoints.

Endpoints with Inequalities

Endpoints in inequalities determine whether the boundary numbers are part of the solution set. With the inequality \(1 \leq x \leq 2\), the symbols \(\leq\) (less than or equal to) and \(\geq\) (greater than or equal to) show that endpoint values are included.

• The presence of \(\leq\) and \(\geq\) means endpoints are included, since "equal to" includes the endpoint in the solution set.
• In interval notation, this inclusion is shown by square brackets: \([1, 2]\).

It is essential to observe the type of inequality symbols used: when endpoints are included, the interval is termed as closed, whereas non-inclusion (regular greater than or less than) results in an open interval, using parentheses, like \((1, 2)\).Grasping the role of endpoints helps in accurately using interval notation and graphically representing a given set on the number line.