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Chapter 0: Problem 69

\(y\) is at least six units from 0 .

Short Answer

Expert verified
The solution for the inequality \(y\) is at least six units from 0 is \(y \leq -6\) or \(y \geq 6\). In other words, \(y\) will be any number less than or equal to -6, and any number greater than or equal to 6.

Step by step solution

01

Interpreting the problem statement

The phrase 鈥榓t least six units away from 0鈥� refers to values either six units to the right of 0 (positive values) or six units to the left of 0 (negative values). This means that our solution includes values that are equal to or more than 6 or equal to or less than -6.
02

Formulating the inequality

So, this problem can be represented as a compound inequality like this: \(y \leq -6\) or \(y \geq 6\). Here, \(y \leq -6\) represents the values six units or more to the left of 0 and \(y \geq 6\) represents the values six units or more to the right of 0.
03

Visualizing the solution on a number line

On the number line, our solution will include all the numbers from -6 marked to the left (including -6) and all the numbers from 6 marked to the right (including 6). Hence, the number line will have a closed circle on -6 and 6 to indicate including those numbers, and arrows pointing to left from -6 and right from 6 up to infinity

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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 straight line that visually represents numbers in order. It鈥檚 like a ruler, but instead of measuring physical length, it is used to represent numbers. Each point on the line corresponds to a number.
Number lines are useful in mathematics because they provide a way to show the position and relationships of numbers, especially when dealing with inequalities or absolute values.
  • Positive numbers are found to the right of zero.
  • Negative numbers are found to the left of zero.
When solving inequalities, number lines help us easily visualize the range and relationship of numbers that satisfy the condition provided by the inequality.
Absolute Value
The concept of absolute value is all about distance. It tells you how far a number is from zero on the number line, regardless of its direction. The absolute value of a number is always positive or zero.
For example:
  • The absolute value of 5 is written as \(|5|\) and equals 5.
  • The absolute value of -5 is written as \(|-5|\) and equals 5 as well.
This is because both 5 and -5 are five units away from zero on the number line. In the problem statement, when we say a number is 'at least six units away from 0,' we mean its absolute value is 6 or more.
Inequalities
Inequalities are mathematical expressions that show how numbers relate to each other in terms of size or order. They don't express exact relationships, but rather define a range of possible solutions.
Some common inequality symbols include:
  • \(>\) means greater than
  • \(<\) means less than
  • \(\geq\) means greater than or equal to
  • \(\leq\) means less than or equal to
In the context of our problem, the inequalities \(y \leq -6\) and \(y \geq 6\) are used to describe numbers that are either greater than or equal to 6, or less than or equal to -6.
Solving Inequalities
Solving inequalities involves finding all possible values that satisfy the inequality condition. In our problem, we are interested in finding numbers that are at least six units away from zero.
Firstly, understand what the inequality is representing. In this case, \(y \leq -6\) or \(y \geq 6\) implies we want numbers outside the interval between -6 and 6.
Next, represent these solutions on a number line to visualize them better.
Include:
  • A closed circle on the boundary numbers to indicate they are part of the solution.
  • Arrows pointing away from these numbers, extending indefinitely to the left for \(y \leq -6\) and to the right for \(y \geq 6\). This shows all numbers less than -6 and greater than 6 are solutions.
This step-by-step approach ensures a clear understanding of inequalities and the set of valid solutions.

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Most popular questions from this chapter

The total numbers \(f\) (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Federal Highway Administration) $$ \begin{array}{|c|c|} \hline 0 \text { Year, } t & \text { Miles traveled, } f(t) \\ \hline 5 & 2423 \\ 6 & 2486 \\ 7 & 2562 \\ 8 & 2632 \\ 9 & 2691 \\ 10 & 2747 \\ 11 & 2797 \\ 12 & 2856 \\ \hline \end{array} $$ (a) Does \(f^{-1}\) exist? (b) If \(f^{-1}\) exists, what does it mean in the context of the problem? (c) If \(f^{-1}\) exists, find \(f^{-1}\) (2632). (d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year was 2747 billion, would \(f^{-1}\) exist? Explain.

Evaluate the indicated function for \(f(x)=x^{2}+1\) and \(g(x)=x-4\) $$(f-g)(3 t)$$

In Exercises 79-84, evaluate the expression for each value of \(x\). (If not possible, state the reason.) \(\frac{x+1}{x-1}\) (a) \(x=1\) (b) \(x=-1\)

\(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) (a) \(\\{(0,1),(1,-2),(2,0),(3,2)\\}\) (b) \(\\{(0,-1),(2,2),(1,-2),(3,0),(1,1)\\}\) (c) \(\\{(0,0),(1,0),(2,0),(3,0)\\}\) (d) \(\\{(0,2),(3,0),(1,1)\\}\)

In Exercises 1-8, find the inverse function of \(f\) informally. Verify that \(f\left(f^{-1}(x)\right)=x\) and \(f^{-1}(f(x))=x\). $$ f(x)=\sqrt[3]{x} $$
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