Problem 8 - Algebra and Trigonometry

Question

7-8 Place the correct symbol \((<,>, \text { or }=)\) in the space. $$ \begin{array}{lllll}{\text { (a) } \frac{2}{3}} & {0.67} & {\text { (b) } \frac{2}{3}} & {-0.67} & {\text { (c) }|0.67|} & {|-0.67|}\end{array} $$

Step-by-Step Solution

Answer

(a) \(=\), (b) \(>\), (c) \(=\)

Step 1: Convert Fractions to Decimals

First, we need to compare the values by converting all fractions to decimals if needed. The fraction \( \frac{2}{3} \) is approximately equal to 0.67 when expressed as a decimal.

Step 2: Compare Decimal Values for Part (a)

We need to compare \( \frac{2}{3} \approx 0.67 \) with 0.67. Since both numbers are equal when expressed as decimals, we place the symbol \( = \) between them. Thus, the correct answer for part (a) is \( \frac{2}{3} = 0.67 \).

Step 3: Compare Values for Part (b)

Here, we compare \( \frac{2}{3} \approx 0.67 \) with -0.67. A positive number (0.67) is always greater than a negative number (-0.67). Hence, \( \frac{2}{3} \) is greater than -0.67, so we use the symbol \( > \). Thus, the correct answer for part (b) is \( \frac{2}{3} > -0.67 \).

Step 4: Compare Absolute Values for Part (c)

Compute the absolute values: \(|0.67| = 0.67\) and \( |-0.67| = 0.67 \). Since both absolute values are equal, we place the symbol \( = \) between them. Therefore, the correct answer for part (c) is \(|0.67| = |-0.67| \).

Key Concepts

Fractions to Decimals

When working with inequalities or comparisons, converting fractions into decimals is often the first step. This conversion helps to easily compare numbers. For example, the fraction \( \frac{2}{3} \) can be converted into a decimal by performing division. By dividing 2 by 3, you’ll find that \( \frac{2}{3} \approx 0.67 \).
This approximation is typically rounded to two decimal places for ease of comparison.
If you deal with other fractions, you could apply the same division method to convert them into decimals.

• Convert fractions to decimals by dividing the numerator by the denominator.
• Round decimals to a convenient number of places, typically two, to maintain consistency across comparisons.

Mastering this conversion process is crucial for making accurate comparisons between numbers in exercises.

Absolute Values

Absolute value is a concept used to describe the distance of a number from zero on a number line, regardless of direction. It essentially strips a number of its sign. For example, both –0.67 and 0.67 have an absolute value of 0.67.
This is written as \(|-0.67| = 0.67\) and \(|0.67| = 0.67\).
Understanding absolute values is important when comparing numbers as it focuses solely on the magnitude.

• Absolute value measures the distance from zero, ignoring the sign.
• Both positive and negative numbers of the same magnitude will have equal absolute values.

Recognizing the absolute values of numbers allows for simpler and more straightforward comparisons especially in math problems that require ignoring the sign.

Decimal Comparison

Once numbers are in decimal form, comparing them becomes straightforward. When comparing two decimal numbers, start with the digits to the left of the decimal point and move right. For instance, when comparing 0.67 and -0.67, start by noticing that 0.67 is positive and greater than any negative number by default.
With decimals like 0.67 compared to exactly 0.67, equivalence is easily recognizable.
In cases of comparison:

• If the decimal points align and numbers to the left are equal, compare rightward from the decimal point.
• Positive decimals always override negative decimals in terms of greater value.

This straightforward tactic aids in ensuring accurate selection of inequality symbols like \(<,>, \text{or} =\). It's crucial in achieving mathematical precision in problems involving decimal comparisons.