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Chapter 2: Problem 11

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

Short Answer

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

Step by step solution

01

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 \(-134\).
02

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 \(-134\).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -3 x+1<-5 \text { or }-4 x-3>-23 $$

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