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Chapter 7: Problem 19

Graph the linear equations and inequalities. $$ z+5>11 $$

Short Answer

Expert verified
Answer: The solution set is all values of \(z\) that are greater than 6, represented as \((6, \infty)\) in interval notation. To represent this on a number line, draw a number line with an open circle at 6 (to indicate that 6 is not included) and shade the region to the right of the open circle to represent all numbers greater than 6.

Step by step solution

01

Simplify the inequality

To simplify the inequality, we need to subtract 5 from both sides: $$ z + 5 - 5 > 11 - 5 \\ z > 6 $$ Now that the inequality is simplified, we can proceed to the next step.
02

Determine the solution set

The inequality states that \(z > 6\). This means the solution set is any number greater than 6. In interval notation, this would be represented as \((6, \infty)\).
03

Graph the solution set

To graph the solution set on a number line: 1. Draw a number line with numbers 0 to 10. 2. Place an open circle at 6 on the number line. An open circle means that the number is not included in the solution set (since \(z > 6\) and not \(z \geq 6\)). 3. Shade the region to the right of the open circle to represent the solution set, which is all numbers greater than 6. Now you have successfully graphed the linear inequality, \(z + 5 > 11\), on a number line.

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Key Concepts

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

Graphing Linear Inequalities
Graphing linear inequalities involves visually representing the set of all solutions on a number line, which helps us understand which values satisfy the inequality. For the inequality \(z + 5 > 11\), simplifying gives us \(z > 6\). Let's break down the graphing process:
  • Draw a number line: Start with a simple horizontal line marked with numbers, typically an evenly spaced range based on the values you are working with. Here, use numbers from 0 to 10.
  • Identify the key point: The inequality \(z > 6\) indicates 6 is crucial. Place an open circle at 6 to show it's not part of the solution (unlike a closed circle, which would mean the point is included).
  • Shade the solution area: For \(z > 6\), shade the line to the right of 6 to indicate all numbers greater than 6 are solutions.
This visual representation is powerful because it immediately communicates the range of values that make the inequality true.
Understanding Solution Sets
A solution set is the collection of all values that satisfy a given inequality or equation. When working with inequalities like \(z > 6\), identifying the solution set involves determining which numbers fulfill the inequality.
For \(z > 6\):
  • Definition: The solution set includes every number that is greater than 6.
  • Real Numbers: Since this inequality involves real numbers, imagine all possible numbers right of 6 on a number line.
  • Visual Tools: Visual aids like number lines and graphs make it easier to comprehend solution sets.
Solution sets are essential in mathematics because they provide a clear understanding of which values meet the conditions of inequalities or equations. By grasping this concept, students can better tackle even more complex problems.
Representing with Interval Notation
Interval notation is a simplified way to express sets of numbers, particularly useful for illustrating solution sets of inequalities. It's a concise language that shows which parts of the number line are included in the set. For the inequality \(z > 6\), the interval notation is written as \( (6, \infty) \).
Here's how it works:
  • Parentheses \(( )\): Used to show that the endpoint is not included in the solution set. For \(z > 6\), 6 is not part of the solution, indicated by a parenthesis.
  • Infinity Symbol \(\infty\): Denotes that the solution set extends indefinitely in a positive direction. Infinity is always represented with a parenthesis because it's not a specific number.
Using interval notation simplifies writing and reading solution sets. It鈥檚 a valuable tool that succinctly communicates which values are valid solutions for an inequality. Understanding and using interval notation can greatly enhance students' mathematical fluency.

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