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Chapter 1: Problem 11

Write the given inequality using interval notation and then graph the interval. $$ 8

Short Answer

Expert verified
Interval notation: \((8, 10]\); graph with open circle at 8 and closed circle at 10.

Step by step solution

01

Analyze the inequality

The inequality is given as \( 8 < x \leq 10 \). This means that \( x \) is greater than 8 and less than or equal to 10.
02

Convert to interval notation

An open interval is used where the boundary is not included, and a closed interval is used where the boundary is included. Since \( x \) is greater than 8 (not including 8), we use \((8,\), and since \( x \) is less than or equal to 10 (including 10), we use \(10]\). Thus, the interval notation is \((8, 10]\).
03

Graph the interval

On a number line, place an open circle at 8 to show that 8 is not included in the interval and a filled circle at 10 to show that 10 is included. Draw a line connecting these two points to represent the numbers between 8 and 10, including 10 but not including 8.

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Key Concepts

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

Interval Notation
Interval notation is a simple way of expressing a range of values that an inequality represents. For the inequality \(8 < x \leq 10\), it shows where \(x\) can exist on the number line.
  • Brackets are used to denote whether endpoints are included or not.
  • An open bracket, \((\) or \()\), means the endpoint is not included.
  • A closed bracket, \([\) or \()]\), signifies the endpoint is included.

In the exercise, since \(x > 8\) and \(x \leq 10\), you write \((8, 10]\) in interval notation. Here, parenthesis \((8,\) denotes 8 is not included, while bracket \(10]\) shows that 10 is included.
Graphing Intervals
Graphing intervals on a number line visualizes the solution set of an inequality. This representation makes it easier to see what parts of the number line are included in the interval.
  • Identify and mark the endpoints of the interval on the number line.
  • If an endpoint is not included, use an open circle.
  • If an endpoint is included, use a filled circle.

For the interval \((8, 10]\), draw an open circle at 8 to indicate it's not part of the solution. Then, draw a filled circle at 10 since it is included. Connect the circles with a line to show all numbers in between.
Number Line Representation
A number line is an effective tool to represent intervals, as it provides a visual depiction of the points included in the range.
  • The number line extends infinitely, but only the relevant segment is considered.
  • Clearly labeled numbers help in accurately plotting endpoints.
  • The line segment and circles allow easy identification of included or excluded points.

For inequalities like \(8 < x \leq 10\), the number line helps you see which numbers are part of the solution. The open circle at 8 means it's not included, and the filled circle at 10 shows it is part of the interval, connected by a horizontal line representing all numbers in between.

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Most popular questions from this chapter

In Problems 1-12, use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{2}-25}{x-5}\) (b) \(\lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5}\)

Discuss what algebra is necessary to evaluate the given limit. Carry out your ideas. $$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{x+27}-3}{x} $$

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{2 x+10}{x^{2}+7 x+10}\) (b) \(\lim _{x \rightarrow-5} \frac{2 x+10}{x^{2}+7 x+10}\)

Use binomial expansion to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{(2 x+1)^{2}-9}{x-1}\) (b) \(\lim _{x \rightarrow 1} \frac{(2 x+1)^{2}-9}{x-1}\)

In Problems \(41-46,\) the given equation is a partial answer to a calculus problem. Solve the equation for the symbol \(y^{\prime}\). $$ 3 y^{2} y^{\prime}-y-x y^{\prime}=x $$
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