Problem 27 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x \leq 0 \text { or } x>10 $$

Step-by-Step Solution

Answer

The interval is \((-\infty, 0] \cup (10, \infty)\).

Step 1: Identify the Inequality

The given inequalities are \( x \leq 0 \) and \( x > 10 \). Our task is to identify what these inequalities represent on a number line.

Step 2: Graph Individual Inequalities

First, graph \( x \leq 0 \). This includes zero and all negative values. On the number line, represent this by shading to the left of zero, including a solid dot at zero.Next, graph \( x > 10 \). This includes all numbers greater than 10. On the number line, represent this by shading to the right of 10 and leaving an open circle at 10.

Step 3: Combine the Graphs

Combine the shaded regions from both inequalities on a single number line. The result will be numbers from negative infinity to zero (inclusive) and numbers greater than 10.

Step 4: Write Interval Notations

Express these ranges in interval notation. \( x \leq 0 \) is written as \((-\infty, 0]\). \( x > 10 \) is written as \((10, \infty)\). Thus, their combined interval notation is \((-\infty, 0] \cup (10, \infty)\).

Key Concepts

Number Line

A number line is a straight, horizontal line that visually represents numbers in increasing order from left to right. Every point on this line corresponds uniquely to a real number. To represent inequalities on a number line, we use different types of markings:

• Solid Dot: This indicates that a number is included in the solution set. For example, in the inequality \(x \leq 0\), zero is part of the solutions, so we mark it with a solid dot.
• Open Circle: This is used to show that a number is not included in the solution. In the inequality \(x > 10\), 10 is not included, so we represent it with an open circle.

Number lines provide a clear visual where you can easily identify the sections that are solutions to the inequalities. For instance, shading to the left of zero, including zero, shows all solutions for \(x \leq 0\). On the other hand, shading to the right of 10 (without including 10) shows solutions for \(x > 10\). The combination of these two shadings gives us a visual representation of all solutions to the compound inequality \(x \leq 0 \text{ or } x > 10\).

Interval Notation

Interval notation is a way of describing sets of numbers between a start and an end point. It's compact and efficient, especially for expressing solutions to inequalities.

• Brackets \([ ]\): Used for inclusive limits, meaning the number at that end is part of the set. For \(x \leq 0\), the notation is \((-\infty, 0]\) because 0 is included.
• Parentheses \(( )\): Used for exclusive limits, which do not include the number at that end. In the case of \(x > 10\), the notation \((10,\infty)\) shows that 10 is not included.
• Union \(\cup\): This symbol indicates the combination of two sets. For instance, the solution \(x \leq 0\) or \(x > 10\) is written as \((-\infty, 0] \cup (10, \infty)\).

Understanding interval notation is crucial for graphing and communicating ranges in math. It's a concise method to describe which numbers make an inequality true.

Graphing

Graphing inequalities involves visually representing their solutions on a number line. It helps us see the range of possible values that satisfy the inequality. Here's how we apply graphing to our example:

• Identify Each Inequality: For \(x \leq 0\), start shading from negative infinity up to and including 0. For \(x > 10\), shade all numbers greater than 10, starting from just above 10.
• Mark Important Points: Use a solid dot at zero because it is included in the solution. Use an open circle at 10 because it is not included.
• Combine Solutions: Once both inequalities are graphed, combine them to show all solutions. The number line will have two distinct shaded areas — one representing numbers less than or equal to zero, and another representing numbers greater than 10.

Graphing gives you a clear visual for inequalities, helping to interpret where solutions overlap or are separate. It's a powerful tool for understanding mathematical relationships, making comparisons easier across different inequality types.