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Chapter 6: Problem 12

Graph each set of real numbers on a number line. \(\\{x \mid-7 \leq x \leq 0\\}\)

Short Answer

Expert verified
To graph the set of real numbers given by the inequality -7 鈮� x 鈮� 0, draw a number line, mark -7 and 0, and highlight or darken the section of the line from -7 to 0.

Step by step solution

01

Identify the boundaries

The given set is defined by the inequality -7 鈮� x 鈮� 0. This tells you that the set begins at -7 and ends at 0. These values are the 'boundaries' of the set.
02

Draw the number line

Draw a straight horizontal line to represent the number line. Put tick marks at regular intervals, and label them with numbers. Be sure to mark -7 and 0. It may be helpful to include a few numbers beyond these boundaries for context.
03

Mark the set on the number line

The set of numbers from -7 to 0 inclusive is represented by a solid line or dot above each number from -7 to 0 on the number line. Since the inequality includes the equal sign (鈮�), this means that -7 and 0 are part of the set, so should be included in your highlighted section.

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Key Concepts

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

Number Line
When dealing with real numbers, the number line is a fundamental visual tool that helps us understand the position and relationship of numbers to one another. It's a straight, horizontal line that goes on infinitely in both directions, representing all possible values from negative infinity to positive infinity.
  • To graph inequalities or sets of real numbers, we use this line as our guide.
  • Each point on the line corresponds to a unique real number.
  • When graphing, it's important to choose a scale and mark off regular intervals to maintain clarity and precision.
  • Numbers are usually placed in ascending order from left to right.
For the inequality \(\{-7 \leq x \leq 0\}\), we simply need to find the points where x equals -7 and 0, and shade or highlight the region between them, including these endpoints.
Real Numbers
Real numbers encompass all the numbers we commonly use, including rational numbers (like fractions and integers) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)). They form a continuous spectrum of values which can be represented on a number line.

In the context of inequalities and graphing on a number line:
  • Inequalities often define a set of real numbers within a specific range.
  • This range can include a variety of numbers, such as whole numbers, decimals, and fractions, as long as they fall within the specified bounds set by the inequality.
  • The term 'inclusive' suggests that the endpoints are part of the set, as is the case with the inequality \(-7 \leq x \leq 0\).
This implies that every real number between -7 and 0, including the endpoints, satisfies the inequality and is part of the set.
Interval Notation
Interval notation is a succinct way to describe the set of numbers that satisfy an inequality. This type of notation uses brackets and parentheses to convey which numbers are included in a set:
Inequality Notation
Inequality notation is used to define the relationship between numbers and show how one number can be less than, greater than, less than or equal to, or greater than or equal to another number.
  • Inequalities are written using symbols such as <, >, 鈮�, and 鈮�.
  • For example, the inequality \(-7 \leq x \leq 0\) tells us that x is a number greater than or equal to -7 and less than or equal to 0.
  • This notation is crucial in both algebraic expressions and graphing because it defines the boundaries of the set of solutions or numbers we are working with.
To interpret and graph this inequality on a number line, we mark the endpoints -7 and 0, ensuring to include them in our highlighted region, as they are part of the solution set due to the 'less than or equal to' sign.

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Most popular questions from this chapter

Solve each equation using the zero-product principle. \((x+11)(x-5)=0\)

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