Problem 4 - Algebra A Combined Function

Question

Graph the solution set of each inequality on a number line and then write it in interval notation. $$ \\{x \mid x<-0.2\\} $$

Step-by-Step Solution

Answer

Interval notation: \((-\infty, -0.2)\).

Step 1: Understanding the Inequality

The inequality \( \{x \mid x < -0.2 \} \) represents all values of \( x \) that are less than -0.2. This means the solution set includes all real numbers that are smaller than -0.2.

Step 2: Graphing the Inequality on a Number Line

To graph \( x < -0.2 \) on a number line, start by locating -0.2. Since \( x \) is less than -0.2 and not equal to it, we use an open circle at -0.2. Shade the region to the left of -0.2 to indicate all numbers less than -0.2.

Step 3: Writing in Interval Notation

In interval notation, values less than -0.2 are written using negative infinity as the lower boundary: \((-\infty, -0.2)\). Here, \(-\infty\) is used to symbolize that there is no lower bound, and it is followed up to but not including -0.2.

Key Concepts

Number Line

A number line is a visual representation of real numbers laid out in a straight line. Each point on this line corresponds to a real number and helps us understand relationships between numbers.
When graphing inequalities, a number line is particularly useful as it provides a clear visualization of which numbers satisfy the inequality. In the case of inequalities like \( x < -0.2 \), we start by identifying the point \( -0.2 \) on the line. Since the inequality does not include \(-0.2\) itself, we draw an open circle at this location.
The open circle indicates that \( x = -0.2 \) is not part of the solution set. All numbers to the left of this open circle, extending infinitely, represent the values that satisfy the inequality \( x < -0.2 \). By shading this entire left side, we effectively communicate the complete solution set of the inequality.

Interval Notation

Interval notation is a concise way to express a range of numbers, especially useful in communicating solutions of inequalities. It uses parentheses \(( )\) and brackets \([ ]\) to denote open and closed ends of an interval, respectively.
For an inequality like \( x < -0.2 \), we write the solution in interval notation as \(( -\infty, -0.2 )\). The parenthesis around \(-0.2\) indicates that this value is not included in the solution set, reflecting the open circle on the number line. On the other hand, \( -\infty \) is always accompanied by a parenthesis, as infinity is not a finite number and cannot be reached, thus it cannot be included.
Interval notation provides an efficient way to communicate the infinite range of numbers which satisfy the condition of the inequality, making it a valuable tool in mathematical expressions.

Real Numbers

Real numbers include all the numbers that exist on the number line, encompassing both rational and irrational numbers. They include integers, fractions, and non-repeating decimals.
The inequality \( x < -0.2 \) deals with real numbers because it includes all rational numbers such as -1, -0.5, and irrational numbers like \(-\sqrt{2}\), as long as they are less than \(-0.2\).
When expressing solution sets using real numbers, it is essential to note that they form a continuum and do not have gaps. Therefore, solutions to inequalities involving real numbers cover a continuous range, extending in the context of \( x < -0.2 \) from \(-0.2\) to negative infinity. This understanding ensures comprehensive interpretation and correct visualization on the number line.