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Chapter 8: Problem 6

Graph each of the following solution sets. A. \((-2,4)\) B. \((-\infty,-2) \cup(3,5]\)

Short Answer

Expert verified
Graph \((-2, 4)\) as an open interval and \((-infty, -2) \cup (3, 5]\) with two separate shaded regions on a number line.

Step by step solution

01

Understanding the Set Notation

To graph solution sets, we need to understand the meaning of interval notation used. The notation \((-2, 4)\) represents all numbers greater than \(-2\) and less than \(4\), excluding \(-2\) and \(4\) themselves (open interval).
02

Graphing the Interval for Set A

On a number line, for the interval \((-2, 4)\), draw an open circle at \(-2\) indicating that \(-2\) is not included, and another open circle at \(4\). Then shade the number line between these two points to represent all numbers in between.
03

Understanding Set B and its Union

The set \((-infty, -2) \cup (3, 5]\) signifies two intervals combined. \((-infty, -2)\) includes all numbers less than \(-2\), without including \(-2\), and \((3, 5]\) includes numbers greater than \(3\) and up to (including) \(5\).
04

Graphing the Intervals for Set B

On a number line, draw an open circle at \(-2\) and shade to the left indefinitely for \((-infty, -2)\). For \((3, 5]\), draw an open circle at \(3\) and a closed circle at \(5\), then shade the number line between \(3\) and \(5\), including \(5\).

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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 mathematical way of writing subsets of the real number line. It helps us express intervals using brackets and parentheses. In this notation:
  • Round parentheses, \'( \' and \' )\', denote that the endpoint is not included in the interval, which is known as an open interval.
  • Square brackets, \'[ \' and \' ]\', mean that the endpoint is included, which is referred to as a closed interval.
For instance, the interval \((-2, 4)\) contains all numbers more than -2 and less than 4, but it doesn鈥檛 include -2 and 4 themselves because both endpoints have round parentheses. Likewise, \((3, 5]\) includes numbers greater than 3 and up to, and including 5, due to the use of a square bracket at the 5.
Number Line
To visualize inequalities and interval notation, a number line is extremely helpful. A number line is a horizontal line that shows numbers in ascending order from left to right. We can use a number line to graphically represent solutions of inequalities and interval notations. Here's how:
  • Draw a horizontal line, marking all integer points as reference.
  • Open circles on the line indicate that numbers at these positions are not included in the set (open interval).
  • Closed circles indicate numbers are included (closed interval).
  • Shade the section between these points to highlight all values included in the interval.
For example, to represent \((-2, 4)\), you draw open circles at -2 and 4, and shade the line between them, indicating all numbers between -2 and 4 are included in the interval.
Open and Closed Intervals
When dealing with intervals, understanding open and closed intervals is crucial. These terms describe whether the endpoints of an interval are included:
  • An open interval, such as \( (a, b) \), excludes its endpoints \(a\) and \(b\). These intervals are denoted with round parentheses, and represented on a graph with open circles.
  • A closed interval, like \([a, b]\), includes both \(a\) and \(b\). Here, square brackets are used, and on a graph, these will be shown with closed or filled-in circles at the endpoints.
Open and closed intervals help precisely describe which numbers belong to a set based on the conditions set by inequalities. For example, in \((3, 5]\), the number 3 is not included, but 5 is part of the set, hence the combination of an open parenthesis and a square bracket.

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Most popular questions from this chapter

What error is made in the following solution? The solutions of \(x^{4}-12 x^{2}+27=0\) are 9 and 3 (SOLUTION CAN'T COPY)

Find a number between 0 and 1 such that the difference of the number and its square is a maximum.

All of the equations we have solved so far have had rational-number coefficients. However, the quadratic formula can be used to solve quadratic equations with irrational or even imaginary coefficients. Solve each equation. $$ 100 i x^{2}+300 x-200 i=0 $$

A table of values for \(f(x)=2 x^{2}-4 x+3\) is shown Explain why it appears that the vertex of the graph of \(f\) is the point \((1,1)\) (IMAGE CANNOT COPY)

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest hundredth. See Using Your Calculator: Solving Quadratic Equations Graphically. $$ x^{2}+x-6=0 $$
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