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Chapter 1: Problem 54

Express the inequality interval notation, and then graph the corresponding interval. $$1 \leq x \leq 2$$

Short Answer

Expert verified
The interval notation is \([1, 2]\) and it includes closed endpoints at 1 and 2.

Step by step solution

01

Identify the Inequality

The given inequality is \(1 \leq x \leq 2\), which states that \(x\) is greater than or equal to 1 and less than or equal to 2.
02

Convert to Interval Notation

For an inequality of the form \(a \leq x \leq b\), the interval notation is \([a, b]\). Therefore, the inequality \(1 \leq x \leq 2\) can be expressed as \([1, 2]\).
03

Graph the Interval

To graph the interval \([1, 2]\), draw a number line. Mark and place a closed circle at 1 and another closed circle at 2, indicating that both endpoints are included. Shade the region on the number line between the two circles inclusive of the endpoints.

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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 to represent a set of numbers within a certain range. It simplifies the expression of inequalities by using brackets and commas. For example, an interval of numbers between 1 and 2, inclusive of both 1 and 2, is written as \([1, 2]\). In this notation:
  • The square bracket \([\) indicates inclusion of the endpoint.
  • The parentheses \(()\) would indicate exclusion, but they are not used here since the endpoints are included.
  • A comma separates the lower and upper bounds of the interval.
This method provides a concise way to display information that would otherwise require a longer description.
Graphing Intervals
Graphing intervals involves representing interval notation visually on a number line. This visual representation can make it easier to understand the range and endpoints of the interval. When graphing an interval like \([1, 2]\):
  • Draw a straight horizontal line, which will serve as your number line.
  • Identify the points you need: in this case, 1 and 2.
  • Place a closed circle on the number line at both 1 and 2 since these numbers are included in the interval.
  • Shade the area between the circles, indicating that all numbers from 1 to 2 are part of the solution set.
Graphing can give a quick, intuitive understanding of which numbers are included in a set. Always ensure that the endpoints are shown correctly, using closed or open circles as required.
Closed Interval
A closed interval is a set of numbers where both endpoints are included in the interval. In interval notation, closed intervals are denoted with square brackets like \([a, b]\). This means every point between and including \(a\) and \(b\) is part of the interval.
  • A closed interval means every value, starting from the lower bound to the upper bound, is included.
  • The usage of closed circles on a number line indicates the inclusion of the endpoints.
  • Closed intervals are handy in defining sets that have definite limits.
In the inequality \(1 \leq x \leq 2\), both endpoint values, 1 and 2, are considered part of the interval. This is why the interval notation is expressed as \([1, 2]\).
Number Line
A number line is a visual tool used in mathematics to represent numbers in a linear format. It helps in understanding the concept of intervals, inequalities, and even basic arithmetic operations. Components of a number line include:
  • A straight horizontal line which is usually divided into equal parts for whole numbers or integers.
  • Points or markers which show specific numbers;
  • Certain segments getting highlighted or marked to denote ranges or intervals.
For the interval \([1, 2]\), the number line will display a segment between the numbers 1 and 2, with this segment being shaded or highlighted. The closed circles at each of these points show they are inclusive. By visualizing number relationships, a number line provides a simple yet powerful means to comprehend numeric concepts.

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Most popular questions from this chapter

Simplify the expression. (a) \(27^{1 / 3}\) (b) \((-8)^{1 / 3}\) (c) \(-\left(\frac{1}{8}\right)^{1 / 3}\)

Simplify the expression. (a) \(\left(\frac{x^{8} y^{-4}}{16 y^{4 / 3}}\right)^{-1 / 4}\) (b) \(\left(\frac{4 s^{3} t^{4}}{s^{2} t^{9 / 2}}\right)^{-1 / 2}\)

Simplify the expression. (a) \(\sqrt{9 a^{3}}+\sqrt{a}\) (b) \(\sqrt{16 x}+\sqrt{x^{5}}\)

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\left(1.062 \times 10^{24}\right)\left(8.61 \times 10^{19}\right)$$

Solve the equation for the variable \(x\). The constants \(a\) and \(b\) represent positive real numbers. $$a^{3} x^{3}+b^{3}=0$$
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