Problem 11 - Elementary Algebra

Question

Graph all solutions on a number line and provide the corresponding interval notation. $$ x \geq-134 $$

Step-by-Step Solution

Answer

Graph: solid dot at -134 with shading to the right. Interval: \([-134, \infty)\).

Step 1: Identify the Inequality

The inequality we are working with is \( x \geq -134 \). This means that \( x \) can be any number that is greater than or equal to \(-134\).

Step 2: Graph the Inequality on the Number Line

To graph this inequality, draw a number line. Locate \(-134\) on this line. Since the inequality is \( x \geq -134 \), you will use a solid dot at \(-134\) to indicate that \(-134\) is included. Then shade the line to the right of \(-134\) to show all numbers greater than \(-134\).

Step 3: Write the Interval Notation

In interval notation, \( x \geq -134 \) is represented as \([-134, \infty)\). The square bracket \([-134\) indicates that \(-134\) is included in the interval, and the parenthesis \(\infty)\) indicates that the interval extends indefinitely to the right.

Key Concepts

Understanding the Number Line

The number line is a powerful visual tool that helps us represent and find solutions to inequalities like \( x \geq -134 \). On a number line, every point corresponds to a real number, and it extends infinitely in both directions. To use it effectively:

• Locate specific numbers using zero as a reference point. For instance, \'\'-134\'\' is to the left of zero.
• Use a solid dot to represent numbers that are included in the solution of the inequality. In this case, because of the \( \geq \) symbol, we place a solid dot at \(-134\).
• Shade the portion of the number line where the inequality holds true. For \( x \geq -134 \), shade to the right of \(-134\), indicating all numbers greater than or equal to \(-134\).

The number line not only helps visualize where the solutions lie but also makes it easier to communicate mathematical ideas clearly.

Interval Notation Explained

Interval notation is a concise way of writing subsets of the real number line. It is especially useful in algebra for expressing inequalities. Consider the expression \( x \geq -134 \). In interval notation:

• The inequality argument starts from \(-134\) which is included in the set, denoted by a square bracket \([-134\).
• Since \( x \) can extend to positive infinity, we express this as \((\infty)\) with a parenthesis acknowledging infinity is not a specific number and can't be attained.
• Combining the two, \( \left[-134, \infty\right) \) means all numbers from \(-134\) to infinity are included, where \(-134\) is part of the solution due to the square bracket.

Understanding interval notation allows for efficient communication and representation of solutions involving infinite sets.

Graphing Inequalities on a Number Line

Graphing inequalities involves visually showing the range of possible solutions on a number line. When dealing with an inequality like \( x \geq -134 \), there are specific steps to follow:

• Start by drawing a straight horizontal line which represents the number line.
• Mark the point \(-134\) clearly on the line.
• Place a solid dot at \(-134\) to indicate that this number is part of the solution set because the inequality is \( \geq \), meaning \'equal to or greater than\'.
• Shade the section of the line that extends to the right of \(-134\), signifying all values that are greater than \(-134\).

This visual representation makes it easy for students to interpret the solutions to inequalities quickly, ensuring a comprehensive understanding of the concept.