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Chapter 1: Problem 71

Decide whether each statement is true or false. \(-|8|>|-9|\)

Short Answer

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True

Step by step solution

01

Understand the Absolute Values

Absolute values represent the distance of a number from zero without considering direction. This means \(|8| = 8\) and \(|-9| = 9\).
02

Apply Absolute Values to the Given Expression

Rewrite the expression \(-|8| > |-9|\)\ using their absolute values calculated in Step 1. This results in \(-8 > -9\).
03

Compare the Values

Compare \(-8\) and \(-9\). As \(-8\) is greater than \(-9\) (because \(-8\) is closer to 0 than \(-9\)), we conclude that the expression is true.

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Key Concepts

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

Understanding Absolute Values
Absolute values measure how far a number is from zero, without considering its direction. Imagine numbers on a number line; the absolute value is just the number's distance from zero. For instance, the absolute value of 8 is 8, and the absolute value of -9 is 9. It doesn't matter if the number is positive or negative 鈥� the absolute value is always positive. When we say \(|8| = 8\), we mean that 8 is precisely 8 units away from zero. Similarly, \(|-9| = 9\) because -9 is 9 units away from zero.
Inequalities Explained
Inequalities are mathematical expressions that show the relationship between two values. They tell us if one number is larger or smaller than the other. For example, the symbols \(>\) and \( <\) indicate 'greater than' and 'less than.' When we have an inequality like \(-|8| > |-9|\), we can use absolute values to simplify this expression to \(-8 > -9|\). Knowing that -8 is closer to zero than -9, we understand that -8 is indeed greater than -9. Thus, the expression is true. This way of thinking can aid in solving various inequality problems.
Using the Number Line
A number line helps visualize the position and distance of numbers from zero. Place zero in the center and plot points for positive and negative numbers on either side. The further a number from zero, the larger its absolute value. By visualizing \(-|8|\) and \(-9|\) on this line, we see that -8 is closer to zero than -9, confirming that \(-8\) is greater than \(-9\). This visual aid simplifies understanding and comparing numbers, especially when dealing with absolute values and inequalities. Always remember, the closer a negative number is to zero, the greater its value.

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Most popular questions from this chapter

Decide whether each statement is true or false. \(|-12|<|-20|\)

Write a numerical expression for each phrase and simplify. The sum of \(-7\) and \(-13,\) increased by 14

Find each difference. $$ 2-(3-5) $$

To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. For example, to find the average of \(14,8,3,9,\) and \(1,\) we add them and then divide by 5. $$ \frac{14+8+3+9+1}{5}=\frac{35}{5}=7 \leftarrow \text { Average } $$ Find the average of each group of numbers. \(18,12,0,-4,\) and \(-10\)

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