Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Tell whether each of the following is true or false. (a) \(-3<-7\) (b) \(-1>-17\) (c) \(-3<-\frac{22}{7}\)

Step-by-Step Solution

Answer

(a) False (b) True (c) False

Step 1: Analyze Inequality (a)

For inequality (a), compare \(-3\) with \(-7\). On the number line, \(-7\) is to the left of \(-3\), which means \(-7\) is less than \(-3\). Thus, the statement \(-3 < -7\) is false.

Step 2: Analyze Inequality (b)

For inequality (b), compare \(-1\) with \(-17\). On the number line, \(-17\) is to the left of \(-1\), meaning \(-1\) is greater than \(-17\). Thus, the statement \(-1 > -17\) is true.

Step 3: Analyze Inequality (c)

For inequality (c), compare \(-3\) with \(-\frac{22}{7}\). First, convert \(-\frac{22}{7}\) to a decimal. \(\frac{22}{7} \approx 3.14\), so \(-\frac{22}{7} \approx -3.14\). On the number line, \(-3.14\) is to the left of \(-3\), meaning \(-3\) is greater than \(-\frac{22}{7}\). Thus, the statement \(-3 < -\frac{22}{7}\) is false.

Key Concepts

Number Line

A number line is an essential mathematical tool that helps us visually compare numbers to each other. It is a straight line where numbers are placed from left to right, with smaller numbers on the left and larger numbers on the right. This representation aids in understanding mathematical concepts, especially inequalities.

In inequalities, knowing the position of numbers on the number line is vital. If one number appears to the left of another, it means it is smaller. Conversely, if it is on the right, it is larger. This concept simplifies the comparison of numbers, whether they are positive or negative.

• Numbers to the right are always greater.
• Numbers to the left are always lesser.

Using the number line helps us easily determine the truth of statements, like proving whether a. a number is greater or lesser than another.
Learning to use it effectively can simplify what might initially seem like complicated comparisons.

Negative Numbers

Negative numbers are numbers less than zero, found on the left side of the number line. A fundamental rule is that the further left a number is, the smaller it is. This can initially be confusing because we often associate bigger numbers with larger values.

When comparing negative numbers, remember:

• A negative number closer to zero is greater than one further from it.
• -1 is greater than -17, since -1 is closer to zero.

Understanding this rule can help clarify inequalities involving negative numbers, such as the equation (b) shown in the exercise.

Visual aids, like number lines, become quite helpful here. They clearly show the positions of numbers and make it obvious why -1 is greater than -17, or why -3 is greater than -7.

Decimal Conversion

Decimal conversion is the process of translating fractions into decimal form. This can ease the comparison of numbers, especially when they are in different formats. It involves dividing the numerator of the fraction by the denominator, giving a decimal result that can easily be placed on the number line.

To compare (\(-\frac{22}{7}\)) with another number, convert it to a decimal first. Performing the division, (\(22 \div 7\)) approximates to 3.14, hence (\(-\frac{22}{7} \approx -3.14\)).
This decimal can then be compared with integers like (-3) by plotting both of them on the number line.

• Placing decimals and integers on a number line helps in easy comparison.
• Seeing them visually demystifies complex-looking inequalities.

Once decimals are lined up, their positions help clarify which is larger, as seen in the conversion done for inequality (c). Thus routine practice and familiarity with decimal conversion can significantly boost numerical comparison skills.