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Chapter 3: Problem 1

The interval (2,5) can be written as the inequality _________________

Short Answer

Expert verified
2 < x < 5

Step by step solution

01

Identify the Interval Type

The interval given is \( (2, 5) \), which represents all numbers between 2 and 5, not including 2 and 5 themselves. This is known as an open interval.
02

Write the Inequality Corresponding to the Interval

An open interval \((a, b)\) corresponds to the inequality \ a < x < b \, where \ x \ represents any number within the interval. For this problem, \(a = 2\) and \(b = 5\) . Therefore, the inequality is \ 2 < x < 5 \.

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Key Concepts

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

Open Interval
An open interval is a way of describing a range of values between two endpoints, where the endpoints themselves are not included in the range. This is visually represented using rounded parentheses like this: \( (a, b) \). For example, in the interval \( (2, 5) \), numbers like 2.1, 3, and 4.9 are included, but the numbers 2 and 5 are not.

This concept is important because it tells us which values are part of the solution set. If both endpoints were included, it would be a **closed interval** and represented with square brackets, like \( [2, 5] \). But for an open interval, only the numbers between the endpoints count.
Inequality Notation
Inequality notation is a mathematical way of expressing the range of values that are included in a set. This notation uses inequality symbols such as **<** (less than) and **>** (greater than) to show the boundaries of the range.

For the open interval \( (2, 5) \), the corresponding inequality notation is written as \[ 2 < x < 5 \]. This means that \ x \ can be any number greater than 2 and less than 5. The inequality tells us exactly where \ x \ can be positioned along the number line.

Understanding how to switch between interval notation and inequality notation is crucial. It helps you to better analyze and solve algebraic problems where you need to find a range of possible solutions.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division). These can be used to represent relationships and solve problems involving intervals.

When we express a range of values (like \( 2 < x < 5 \)) as an inequality, we鈥檙e actually working with an algebraic expression. The variables and the inequality signs together tell us about the range of solutions.

For example, if you solve an equation and find the solutions lie between two values, you鈥檒l often express this in both interval and inequality notation. If you know how to translate between these notations, it will be easier to understand and solve more complex math problems.

Practice converting between different notations and solving related equations to strengthen your grasp of these algebraic concepts.

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Most popular questions from this chapter

\(G(x)=-x^{4}+32 x^{2}+144\) (a) Determine whether \(G\) is even, odd, or neither. (b) There is a local maximum value of 400 at \(x=4\). Find a second local maximum value. (c) Suppose the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=0\) and \(x=6\) is 1612.8 square units. Using the result from (a), determine the area of the region enclosed by the graph of \(G\) and the \(x\) -axis between \(x=-6\) and \(x=0\).

Which of the following functions has a graph that is symmetric about the \(y\) -axis? (a) \(y=\sqrt{x}\) (b) \(y=|x|\) (c) \(y=x^{3}\) (d) \(y=\frac{1}{x}\)

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, \(y=x^{2}\) ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. $$ g(x)=3|x+1|-3 $$

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=2 x^{2}-3 x+1\)

Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: (0,-3) and (3,0)\(;\) a local maximum value of -2 at \(-1 ;\) a local minimum value of -6 at \(2 .\) Compare your graph with those of others. Comment on any differences.
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