Problem 52 - Precalculus Mathematics for Calculus

Question

Express the inequality in interval notation, and then graph the corresponding interval. $$1 \leq x \leq 2$$

Step-by-Step Solution

Answer

The inequality \(1 \leq x \leq 2\) is expressed as \([1, 2]\) and graphed with solid dots at 1 and 2.

Step 1: Understand the Inequality

The inequality given is \(1 \leq x \leq 2\). This means that \(x\) can be any number between 1 and 2, including the numbers 1 and 2 themselves. It's a closed interval because both endpoints are included.

Step 2: Translate to Interval Notation

A closed interval that includes both endpoints is written with square brackets. Thus, the inequality \(1 \leq x \leq 2\) is expressed in interval notation as \([1, 2]\).

Step 3: Graph the Interval on the Number Line

To graph the interval \([1, 2]\):- Draw a number line.- Mark the numbers 1 and 2 on the line.- Place a filled (solid) dot at both 1 and 2 to indicate that these endpoints are included. - Shade the line segment between 1 and 2 to represent all numbers in this interval.

Key Concepts

Number Line

A number line is a simple yet powerful tool to visually represent numbers and intervals on a straight line. It allows us to see the position of different numbers relative to each other. Imagine a horizontal line extending infinitely in both directions. This line has numbers marked off at equal distances:

• The center is usually marked as zero.
• To the right are positive numbers.
• To the left are negative numbers.

For inequalities like the one given, a number line helps visualize the range of possible solutions. Here, the endpoints of the interval are the most significant parts. You mark these endpoints precisely to include or exclude them, depending on the type of interval. In our example, because we use a closed interval, filled dots at 1 and 2 show these numbers are included in the solution.

Inequalities

Inequalities are mathematical expressions that involve the symbols ">", "<", "≤", or "≥". They are used to compare values and represent a range of possible solutions rather than a single number.

• The symbol "≤" (less than or equal to) shows that one number is smaller or equal to another.
• Likewise, the symbol "≥" (greater than or equal to) indicates that a number is larger or equal to another.

In expressions like 1 ≤ x ≤ 2, inequalities define a set of numbers that satisfy the conditions.

This specific notation tells us "x" can be any number from 1 to 2, including those endpoints. Understanding inequalities is crucial for distinguishing different types of intervals, whether open or closed.

Closed Interval

A closed interval is defined using square brackets, such as \[ [a, b] \]. This indicates that all numbers from "a" to "b", including the numbers "a" and "b", are part of the interval. Closed intervals are used when endpoints are part of the solution set.

Key features of closed intervals are:

• Include both endpoints "a" and "b".
• Graphically represented with solid dots on a number line.

In the example \( 1 \leq x \leq 2 \) translated into closed interval notation,\[ [1, 2] \], it asks us to consider all numbers including 1 and 2. Representation of closed intervals is crucial for accurately graphing and understanding the entirety of available solutions in a set.