Problem 34 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x \leq 0 \text { or } x>0 $$

Step-by-Step Solution

Answer

The solution is the entire number line except for 0, in interval notation: \((- fty, 0] \cup (0, \infty)\).

Step 1: Understand the Inequalities

The compound inequality given is \( x \leq 0 \) or \( x > 0 \). This statement includes all numbers that are either less than or equal to 0 or greater than 0.

Step 2: Graph the Solutions on a Number Line

For \( x \leq 0 \), shade the region from negative infinity up to and including 0. Draw a closed circle at 0 to indicate that 0 is included. For \( x > 0 \), shade the region from 0 to positive infinity, using an open circle at 0 to indicate that 0 is not included.

Step 3: Write the Interval Notation

In interval notation, \( x \leq 0 \) is represented as \((-fty, 0]\) and \( x > 0 \) as \((0, \infty)\). The union of these two intervals, which represents the entire number line without any gaps, is \((-fty, 0] \cup (0, \infty)\).

Key Concepts

Number Line

A number line is a straight line that visually represents numbers in order, spaced equally apart. It allows us to easily see and understand the properties of numbers, especially when dealing with inequalities.

When you solve inequalities or graph solutions, the number line becomes particularly useful. For example, you can graph a simple inequality like \(x \leq 0\) by shading the segment of the number line from negative infinity to 0 and drawing a closed circle at 0. This closed circle means that 0 is included in the solution.

• Open circles indicate that the number is not part of the solution.
• Closed circles indicate inclusion of the number in the solution.

This visual representation helps to make the abstract concept of inequalities more tangible. With the number line, you easily show which numbers satisfy the inequality.

Interval Notation

Interval notation offers a compact way to express parts of a number line, making it easy to show where inequalities hold true.

Let's break down how it works:

• An interval such as \((-\infty, 0]\) includes all numbers from negative infinity up to and including 0. The bracket \([\) indicates inclusion of the endpoint.
• The interval \((0, \infty)\) includes all numbers greater than 0, but does not include 0. The curved parenthesis \(()\) indicates exclusion of the endpoint.

Combining intervals using the union symbol \(\cup\) shows all numbers that belong to either interval. For example, in the given inequality, \((-\infty, 0] \cup (0, \infty)\) represents the entire number line, since every real number either is less than or equal to 0 or greater than 0. This way, interval notation provides a precise and readable format.

Compound Inequalities

Compound inequalities involve two or more simple inequalities combined by the words "and" or "or." Understanding their meaning is crucial for solving and graphing solutions correctly.

When using "or" as in \(x \leq 0\) or \(x > 0\), the solution includes values that satisfy at least one of the inequalities. Essentially, any number from negative infinity to positive infinity meets the condition because every real number is either less than or equal to zero or greater than zero.

• "Or" combines solutions, expanding the solution set.
• "And" limits the solution set to only those numbers satisfying both conditions simultaneously.

Knowing this difference helps you correctly interpret compound inequalities and express their solutions either graphically on a number line or symbolically with interval notation.