Problem 26 - Intermediate Algebra

Question

Graph each inequality. $$ x \geq 0 $$

Step-by-Step Solution

Answer

Draw a number line with a closed circle at 0 and an arrow extending to the right.

Step 1: Understanding the Inequality

The inequality is given as \( x \geq 0 \). This expression indicates that \( x \) can take any real number value that is greater than or equal to \( 0 \). This means \( x \) includes values from \( 0 \) to infinity.

Step 2: Drawing the Number Line

To graph the inequality \( x \geq 0 \), we start by drawing a horizontal number line. Mark \( 0 \) on the line, and also indicate numbers such as \( 1, 2, -1, -2 \), and so on to represent a portion of the real line.

Step 3: Showing the Inequality with Symbols

Since the inequality is \( x \geq 0 \), we place a closed circle or filled dot at \( 0 \) on the number line to indicate that \( 0 \) is included as a value of \( x \).

Step 4: Indicating the Solution Set

To represent all values for \( x \geq 0 \), draw an arrow starting from the closed circle at \( 0 \) extending to the right, indicating all numbers greater than \( 0 \). The arrow should continue to the right, representing that it goes on indefinitely, showing all numbers greater than 0.

Key Concepts

Number Line

A number line is a visual tool used to represent numbers in a linear fashion. It helps to easily understand and interpret mathematical concepts like inequalities. To create a number line, draw a straight horizontal line using a ruler.

• On this line, select a point to represent the number zero (0).
• Mark equal intervals on both sides of zero to represent positive and negative numbers, like "1, 2, 3" and "-1, -2, -3."
• These marks can extend infinitely in both directions. You can show this infinity by placing arrows on both ends of the line.

For graphing inequalities like \( x \geq 0 \), using a number line helps to visually show all possible solutions. It makes it straightforward to identify the range of numbers that satisfy the inequality and to determine what number sets are involved.

Solution Set

The term "solution set" refers to all possible values that satisfy a given inequality statement. For the inequality \( x \geq 0 \), the solution set includes all numbers from zero and beyond.

• In mathematics, solution sets for inequalities might be represented in interval notation. For \( x \geq 0 \), this notation would be \([0, \infty)\).
• This means that zero is part of the solution (closed bracket), and the numbers extend indefinitely towards infinity in the positive direction (open bracket).

When graphing the solution set on a number line, it is shown by a closed circle on zero and an arrow pointing right. This visual translates the written inequality into an easily understood representation, where the closed circle signals inclusion of the starting point zero.

Inequality Symbols

Inequality symbols are crucial for understanding and expressing relationships between numbers. They show how values compare and indicate which numbers can satisfy an equation.

• For instance, the symbol \( \geq \) stands for "greater than or equal to."
• In a graph of \( x \geq 0 \), it means including all values equal to or larger than zero.
• The \( > \) and \( < \) symbols indicate a strict inequality, meaning values cannot equal the compared number. In contrast, \( \geq \) and \( \leq \) allow for equality.

By using inequality symbols, mathematicians can precisely describe solution sets and convey complex ideas succinctly. In practical uses, matching symbols correctly with their visual representation on a number line brings clarity to mathematical statements, making them accessible even to beginner learners.