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Chapter 5: Problem 234

Order each pair of numbers, using \(<\) or \(>\) \(-\frac{2}{3}\) ____ \(-0.632\)

Short Answer

Expert verified
-\frac{2}{3} < -0.632

Step by step solution

01

Understand the Task

You need to determine if \( -\frac{2}{3} \) is less than or greater than \( -0.632 \) by using the symbols \( < \) or \( > \).
02

Convert Fractions to Decimals

Convert the fraction to a decimal to compare it more easily. \( -\frac{2}{3} \) can be converted to a decimal by performing the division \( -2 \div 3 = -0.6667 \).
03

Compare the Numbers

Compare the decimal equivalents. You have \( -0.6667 \) and \( -0.632 \). Since \( -0.6667 \) is less than \( -0.632 \), \( -0.6667 < -0.632 \).
04

Write the Inequality

Substitute the original numbers back into the inequality, giving \( -\frac{2}{3} < -0.632 \).

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Key Concepts

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

inequality symbols
Inequality symbols are like little signs that show us if one number is bigger or smaller than another number. There are mainly two such symbols, < and >.
The symbol < means 'less than'. It points to the smaller number. For example, in '3 < 5', 3 is less than 5.
The symbol > means 'greater than'. It points to the bigger number. For example, in '7 > 4', 7 is greater than 4.
Remembering which way the symbols point is crucial. One simple trick is to imagine the symbol < or > talking; it always wants to eat the bigger number, so its 'mouth' is open towards the larger value!
Once you master these symbol meanings, comparing numbers becomes much easier.
converting fractions to decimals
Sometimes it is hard to compare a fraction like \( -\frac{2}{3} \) with a decimal like -0.632 directly. To make it easier, we can convert the fraction to a decimal. This means turning \( \frac{2}{3} \) into a form with digits after a decimal point.
To convert a fraction to a decimal, you divide the top number (numerator) by the bottom number (denominator). So for \( \frac{2}{3} \), you do the division \( 2 \div 3 = 0.6667 \). Don't forget to keep the negative sign if the number is negative!
Now, instead of comparing \( -\frac{2}{3} \) and -0.632, we can compare -0.6667 and -0.632. Visual comparison becomes much easier, allowing us to use inequality symbols correctly.
number comparison
Comparing numbers, whether they are fractions or decimals, is a fundamental math skill. After converting fractions to decimals, the next step is to compare their sizes.
Let's compare -0.6667 and -0.632. It's important to remember that with negative numbers, the number that looks 'bigger' in absolute value is actually smaller.
In this case, -0.6667 is further left on the number line than -0.632, so \( -0.6667 < -0.632 \).
Finally, substitute your original numbers back into the comparison: \( -\frac{2}{3} < -0.632 \). This process makes sure you are comparing the right values and using the inequality symbols correctly to write the answer.

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