Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Give a verbal description of the subset of real numbers that is represented by the inequality, and sketch the subset on the real number line.\(0

Step-by-Step Solution

Answer

The inequality \(0

Step 1: Interpret the Inequality

The inequality \(0< x \leq 6\) can be read as: 'All numbers greater than 0 and less than or equal to 6'. This implies that the number x is an exclusive subset of all real numbers which is greater than zero (0) but less than or equal to six (6). Zero is not included in the subset as indicated by the less than symbol ('<'). However, six is included as indicated by the less than or equal to symbol ('\leq').

Step 2: Plot on the Number Line

To plot this on a real number line, a line with evenly spaced marks indicating the real numbers is drawn. Zero and six are specifically indicated, as they are the boundaries of the subset. An open circle is then marked at zero to show that zero isn't included in the subset and a closed circle is marked at six to show that six is included. Then, a shaded line or arrow is drawn from zero (exclusive) to six (inclusive) representing the numbers that are part of this subset.

Key Concepts

Real Number Subset

In mathematics, a subset refers to a portion of a larger set. Here, we focus on subsets of real numbers. Real numbers include all the numbers on the number line, which encompasses both rational and irrational numbers. When we talk about subsets of real numbers, we're referring to a specific collection of numbers that meet certain criteria.

For the inequality given, the subset is all the real numbers greater than 0 but less than or equal to 6. This subset includes numbers like 0.1, 2.5, and 6 but does not include 0 itself. The concept of subsets is vital because it allows us to precisely define and work with specific parts of the number line rather than handling all real numbers at once. Understanding the idea of subsets helps in solving problems accurately as it clarifies which numbers are included in the solution and which are not.

Number Line Representation

Number lines are visual tools that allow us to display real numbers and their relationships concisely. They help in understanding inequalities and subsets. For the given inequality, the number line will represent real numbers from 0 to 6.

Start by drawing a horizontal line with evenly spaced intervals. This line represents the entire set of real numbers. Mark special points like 0 and 6, the boundaries of the given subset, prominently. To show that 0 is not included in the subset, draw an open circle at 0. For 6, draw a closed circle to indicate its inclusion.

• Open Circle: Indicates a number is not part of the subset.
• Closed Circle: Indicates a number is part of the subset.

Then, shade the line between these two points to illustrate the numbers that belong to the subset from (not including) 0 to 6 (including). This visual representation makes understanding and comparing numbers in the subset much clearer.

Less Than and Greater Than Concepts

Inequalities use the concepts of 'less than' and 'greater than' to describe relationships between numbers. These symbols help identify which numbers are part of a subset. In the given inequality,

\(0 < x \leq 6\),

• '<' (less than) means numbers more than 0 but not including 0 itself are in the subset.
• '\(\leq\)' (less than or equal to) means numbers less than or equal to 6, including 6, are in the subset.

Grasping these concepts is crucial because they direct us in accurately identifying the portions of the number line relevant to our subset. This notation is universal in expressing mathematical ranges and boundaries, reducing complexities and helping streamline problem-solving involving real numbers.

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Other exercises in this chapter

Problem 36

Simplify the expression.\(\left(\frac{5}{z}\right)^{2}\left(\frac{2}{z}\right)^{3}\)

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Problem 36

Identify the rule(s) of algebra illustrated by the statement.\(x+(y+10)=(x+y)+10\)

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Problem 36

Perform the indicated operations and simplify.\(\frac{x+13}{x^{3}(3-x)} \cdot \frac{x(x-3)}{5}\)

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Problem 36

Factor the trinomial.\(12 y^{2}+7 y+1\)

View solution

Key Concepts

Real Number Subset

In mathematics, a subset refers to a portion of a larger set. Here, we focus on subsets of real numbers. Real numbers include all the numbers on the number line, which encompasses both rational and irrational numbers. When we talk about subsets of real numbers, we're referring to a specific collection of numbers that meet certain criteria.

For the inequality given, the subset is all the real numbers greater than 0 but less than or equal to 6. This subset includes numbers like 0.1, 2.5, and 6 but does not include 0 itself. The concept of subsets is vital because it allows us to precisely define and work with specific parts of the number line rather than handling all real numbers at once. Understanding the idea of subsets helps in solving problems accurately as it clarifies which numbers are included in the solution and which are not.

Number Line Representation

Number lines are visual tools that allow us to display real numbers and their relationships concisely. They help in understanding inequalities and subsets. For the given inequality, the number line will represent real numbers from 0 to 6.

Start by drawing a horizontal line with evenly spaced intervals. This line represents the entire set of real numbers. Mark special points like 0 and 6, the boundaries of the given subset, prominently. To show that 0 is not included in the subset, draw an open circle at 0. For 6, draw a closed circle to indicate its inclusion.

• Open Circle: Indicates a number is not part of the subset.
• Closed Circle: Indicates a number is part of the subset.

Then, shade the line between these two points to illustrate the numbers that belong to the subset from (not including) 0 to 6 (including). This visual representation makes understanding and comparing numbers in the subset much clearer.

Less Than and Greater Than Concepts

Inequalities use the concepts of 'less than' and 'greater than' to describe relationships between numbers. These symbols help identify which numbers are part of a subset. In the given inequality,

\(0 < x \leq 6\),

• '<' (less than) means numbers more than 0 but not including 0 itself are in the subset.
• '\(\leq\)' (less than or equal to) means numbers less than or equal to 6, including 6, are in the subset.

Grasping these concepts is crucial because they direct us in accurately identifying the portions of the number line relevant to our subset. This notation is universal in expressing mathematical ranges and boundaries, reducing complexities and helping streamline problem-solving involving real numbers.