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Chapter 0: Problem 19

In Exercises 19-30, (a) give a verbal description of the subset of real numbers represented by the inequality or the interval, (b) sketch the subset on the real number line, and (c) state whether the interval is bounded or unbounded. $$ x \leq 5 $$

Short Answer

Expert verified
The inequality \(x \leq 5\) represents all real numbers less than or equal to 5. On a number line, this subset extends from 5 (inclusive) to negative infinity, which makes it unbounded.

Step by step solution

01

Describe the Subset of Real Numbers

The inequality \(x \leq 5\) describes all real numbers \(x\) that are less than or equal to 5.
02

Sketch the Subset on The Real Number Line

On a real number line, the set of numbers satisfying \(x \leq 5\) would include 5 (represented by a filled-in dot) and extend to the left of 5, covering all numbers less than 5.
03

Determine if the Interval is Bounded or Unbounded

The interval is bounded from above by 5, but there is no lower limit, meaning it extends indefinitely toward negative infinity. Therefore the interval is considered unbounded.

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Key Concepts

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

Describing Inequalities
When working with inequalities in trigonometry and other branches of mathematics, it's essential to understand their basic structure and what they represent. Inequalities like \(x \leq 5\) tell us about the relative size or order of numbers. This particular inequality means that the value of \(x\) is less than or equal to 5.

In other words, \(x\) could be 5, 4, 3, or any other number down to negative infinity. This is a way to describe a range of numbers rather than a single solution, which is often the case with an equation involving an equals sign. Inequalities can be strict (using < and >) or include equality (using \(\leq\) and \(\geq\)). Describing the subset of real numbers helps us to understand the scope and limits of possible values of \(x\) within a specific context.
Real Number Line Representation
The real number line is a visual tool to represent numbers, allowing us to place any real number on it, usually drawn horizontally with positive numbers to the right of zero and negative numbers to the left. Inequalities are often best understood when represented on a number line.

For the inequality \(x \leq 5\), we demonstrate this graphically by drawing a line that extends from 5 to negative infinity. A filled-in circle at 5 indicates that 5 is a part of the solution set. This graphic representation lets us quickly and easily see which numbers are included in the inequality, aiding not just in understanding but also in problem-solving.
Bounded and Unbounded Intervals
When discussing intervals on a number line, we often classify them as 'bounded' or 'unbounded' to describe their limits. An interval is bounded when it has both a lower and upper limit. For instance, the interval \([2, 6]\) is bounded because it only includes numbers between 2 and 6.

In contrast, the interval described by \(x \leq 5\) is unbounded, as it continues infinitely to the left on the number line. While it has an upper limit (5), it lacks a lower limit, signifying that it is unbounded at the bottom. Understanding whether an interval is bounded or unbounded helps in recognizing the extent of the solution set and in applying the appropriate mathematical concepts when solving problems.

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Most popular questions from this chapter

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ \(f(x)=-\sqrt{x-2}+5 &\quad x_{1}=3, x_{2}=11\) \end{array} $$

The numbers of households \(f\) (in thousands) in the United States from 1995 to 2003 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Census Bureau) $$ \begin{array}{|c|c|} \hline \text { Year, } t & \text { Households, } f(t) \\ \hline 5 & 98,990 \\ 6 & 99,627 \\ 7 & 101,018 \\ 8 & 102,528 \\ 9 & 103,874 \\ 10 & 104,705 \\ 11 & 108,209 \\ 12 & 109,297 \\ 13 & 111,278 \\ \hline \end{array} $$ (a) Find \(f^{-1}(108,209)\). (b) What does \(f^{-1}\) mean in the context of the problem? (c) Use the regression feature of a graphing utility to find a linear model for the data, \(y=m x+b\). (Round \(m\) and \(b\) to two decimal places.) (d) Algebraically find the inverse function of the linear model in part (c). (e) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(117,022)\). (f) Use the inverse function of the linear model you found in part (d) to approximate \(f^{-1}(108,209)\). How does this value compare with the original data shown in the table?

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=-\frac{2}{x} $$

Maximum Profit The cost per unit in the production of a portable CD player is \(\$ 60\). The manufacturer charges \(\$ 90\) per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by \(\$ 0.15\) per CD player for each unit ordered in excess of 100 (for example, there would be a charge of \(\$ 87\) per CD player for an order size of 120 ). (a) The table shows the profit \(P\) (in dollars) for various numbers of units ordered, \(x\). Use the table to estimate the maximum profit. \begin{tabular}{|l|c|c|c|c|} \hline Units, \(x\) & 110 & 120 & 130 & 140 \\ \hline Profit, \(P\) & 3135 & 3240 & 3315 & 3360 \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|} \hline Units, \(x\) & 150 & 160 & 170 \\ \hline Profit, \(P\) & 3375 & 3360 & 3315 \\ \hline \end{tabular} (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x\) ? (c) If \(P\) is a function of \(x\), write the function and determine its domain.

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. Proof Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.
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