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Chapter 14: Problem 47

Graph the set. (GRAPH CANNOT COPY) $$\\{x | x \leq-4\\}$$

Short Answer

Expert verified
The graph representing the set of numbers \(x\) where \(x \leq -4\) includes the point at -4 and every point to its left on the number line. The points to the left represent the set of all numbers less than -4. The point at -4 is normally represented by a filled dot, with the arrow pointing to the left representing the numbers less than -4.

Step by step solution

01

Understand Inequations

In an equation, an inequation defines a part of the number line that contains an infinite number of values. Here, we have \(x \leq -4\), where \(x\) can take any value that is less than or equal to -4.
02

Draw the number line

The number line is an illustrative tool where points or segments of the line represent different numbers on the line. Draw a number line to start and place -4 in the middle. Mark off points to the right and left of -4 with greater and smaller numbers respectively.
03

Graph the inequality

Since the exercise is asking to graph all numbers where \(x\) is less than or equal to -4, we go to the point at -4 and darken it to show it is included, due to the 'or equals to' part of the inequality \(\leq\). Then, draw an arrow to the left, indicating all the numbers less than -4.

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Key Concepts

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

Number Line
A number line is a straightforward graphical representation used in math, especially in the context of inequalities. It is a straight horizontal line where numbers are placed at intervals. To better understand the number line, visualize it as a timeline with points representing numbers. Here is how you use it:
  • Draw a horizontal line.
  • Mark equal spaces to represent increments of numbers.
  • Choose a point to represent specific numbers, such as integers.
  • Typically, smaller numbers lie to the left and larger numbers to the right.
Each point on the number line represents a number's exact location. In our case, -4 is a crucial point, because it serves as a boundary in our inequality problem. This makes the number line essential for visually solving and understanding inequalities.
Inequality Representation
Inequality representation is a tool to express that one value is larger, smaller, or equal to another. Inequalities are typically denoted using signs such as \(<, >, \leq, \geq \). In this exercise, we deal with the specific inequality:
  • \(x \leq -4\)
This means that \(x\) can be any value equal to or less than -4. It's important to remember:
  • \(\leq\) implies that the value is inclusive鈥攊n this case, -4 is part of the solution.
  • The solution will include values continuing infinitely in the direction suggested by the inequality.
Inequality representation is key in many areas of math, where understanding the range of solutions is necessary. On a number line, such representation often involves an arrow from the boundary point, demonstrating the continuum of values.
Prealgebra Concepts
When diving into inequalities and number lines, you engage with prealgebra concepts foundational for future math courses. Prealgebra focuses on:
  • Basic arithmetic operations.
  • Understanding integers, decimals, and fractions.
  • Comprehending math symbols and their meanings.
  • Grasping ideas like negative numbers and their placement on the number line.
In our specific problem, concepts of graphing simple inequalities are front and center. This exercise requires an understanding of:
  • What it means for a solution to be less than or equal to a number.
  • How to express solutions on a number line.
These prealgebra topics are crucial for young learners, as they form the base for algebra and more advanced math concepts.

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