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Chapter 9: Problem 78

True or false? If } a<0, \text { then }|a|=-a

Short Answer

Expert verified
True

Step by step solution

01

Understanding the Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. Thus, for any real number \( a \), \( |a| \) is always non-negative.
02

Applying the Condition \( a

Given \( a<0 \), the number is negative. For negative numbers, the absolute value is taken by negating the number. Therefore, \( |a| = -a \) when \( a<0 \).
03

Conclusion

Since the statement holds true under the given condition \( a<0 \), we conclude that the statement is true.

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Key Concepts

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

negative numbers
Negative numbers are numbers that are less than zero. They are found to the left of zero on the number line. These numbers are often indicated by a minus sign (鈭�) in front of them. For example, 鈭�5, 鈭�2, and 鈭�0.75 are all negative numbers. Understanding negative numbers is important because they show up in a variety of real-life scenarios, like temperatures below freezing and debts. When dealing with negative numbers in calculations, it鈥檚 essential to know the rule of signs: when multiplying or dividing two negative numbers, the result is positive; but when adding or subtracting, it depends on the operation done on the absolute values of the numbers.
number line
A number line is a visual representation of numbers on a straight line, which helps in understanding their order and relationship. Zero is the central point, with positive numbers to the right and negative numbers to the left. Each point on the number line corresponds to a unique number. This makes it easier to compare the size of numbers and to perform operations like addition and subtraction. For instance, moving to the right on the number line represents adding a positive number, whereas moving to the left represents adding a negative number or subtracting a positive one.
distance from zero
The distance from zero on a number line is known as the absolute value of a number. It signifies how far a number is from zero without considering its direction. For any number a, the absolute value is denoted as |a|. The interesting thing is, whether a number is positive or negative, its absolute value is always non-negative. For example, |3| = 3 and |鈭�3| = 3. This is because distance is always non-negative. To find the absolute value, simply look at the number and ignore the sign, if any. For negative numbers, you can think of it as flipping the sign to make it positive. This concept is very useful in various fields of mathematics and real-world applications.

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Most popular questions from this chapter

Explain how to determine whether to use a dashed line or a solid line when graphing a linear inequality in two variables.

For each compound inequality, give the solution set in both interval and graph form. $$ -3 x<3 \text { and } x+2<6 $$

Express each set in the simplest interval form. (Hint: Graph each set and look for the intersection or union. $$ [-1, \infty) \cap(-\infty, 9] $$

Express each set in the simplest interval form. (Hint: Graph each set and look for the intersection or union. $$ [-1,2] \cup(0,5) $$

Evaluate. $$ (-5)-|-9|+|5-4| $$
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