1. Suppose a and b are integers, and the absolute value of a is > than the absolute value of b. Must a be greater than b? Use examples to support your answer.
2. Do decimals have opposites? Explain.
3. For what values of x does the absolute value of x equal negative x?
#1.
Not necessarily.
For example, a = -5 and b = -3
|a| = 5
|b| = 3
|a| is greater than |b|, but
-5 is NOT greater than -3.
#2.
Opposites are numbers which have the same distance (numerical value) from zero but different in directions (opposite signs). An example would be -5 and 5.
I'm not really sure about this, but I think decimals have opposites, since they can be positive or negative.
#3.
|x| = -x, or
|x| + x = 0
Note that this is true for all x less than or equal to zero.
Hope this helps~ :3
14-12*3
1. No, a does not have to be greater than b. The absolute value only measures the distance of a number from zero on the number line, and does not consider the sign of the number. Therefore, it's possible for a smaller number to have a larger absolute value than a bigger number. Let's take some examples to illustrate this:
Example 1: a = -5 and b = 7
|a| = |-5| = 5
|b| = |7| = 7
Here, the absolute value of a (5) is less than the absolute value of b (7), but a is still smaller than b.
Example 2: a = -10 and b = -3
|a| = |-10| = 10
|b| = |-3| = 3
In this case, the absolute value of a (10) is greater than the absolute value of b (3), and a is also greater than b.
So, the absolute value alone cannot determine the relationship between a and b.
2. Decimals do have opposites. The opposite of a decimal is the same number, but with a different sign. In other words, if you change the positive sign to negative, or the negative sign to positive, you will get the opposite of the decimal.
For example:
The opposite of 3.2 is -3.2.
The opposite of -5.7 is 5.7.
The opposite of 0.9 is -0.9.
So, decimals have opposites just like whole numbers and fractions.
3. The absolute value of x equaling negative x is only possible when x is equal to 0.
The absolute value of a number is always greater than or equal to zero. It represents the distance of the number from zero on the number line. The negative of a number is always less than or equal to zero.
So, if the absolute value of x equals negative x, it means that both values are equal to zero. In other words, the number x must be equal to zero for the equation to hold.
Therefore, the only value of x for which the absolute value of x equals negative x is x = 0.
1. To determine whether a must be greater than b when the absolute value of a is larger than the absolute value of b, we can start by understanding what absolute value represents. The absolute value of a number is its distance from zero on the number line. Therefore, it is always a non-negative value.
Now, let's consider a few examples:
Example 1: a = 5, b = -3
In this case, both absolute values are 5 and 3, respectively. Since 5 is greater than 3, we can conclude that a is greater than b.
Example 2: a = -2, b = -7
Here, the absolute value of a is 2, and the absolute value of b is 7. Again, we see that 7 is greater than 2. Thus, a is not necessarily greater than b.
From these examples, it is clear that a does not have to be greater than b when the absolute value of a is larger than the absolute value of b.
2. Decimals can indeed have opposites. The opposite of a decimal is the number that, when added to the original decimal, results in zero. To find the opposite of a decimal, we change its sign while keeping the same magnitude.
For example, let's consider the decimal -0.8. Its opposite would be 0.8 because -0.8 + 0.8 equals zero. Similarly, the opposite of 2.3 would be -2.3, and so on.
The concept of opposites extends to all real numbers, whether they are whole numbers, fractions, or decimals. It allows us to describe numbers symmetrically on the number line, where the opposite values are equidistant from zero.
3. The absolute value of a number, denoted by |x|, is always non-negative. Therefore, to have the absolute value of x equal to negative x, the negative of x must itself be non-negative. In other words, x must be equal to 0.
To see this, let's apply the definition of absolute value: |x| = -x. Since -x is non-negative, it implies that -x is either zero or a positive number. However, if -x were positive, then |x| = -x would also be positive, contradicting the initial assumption. Therefore, the only solution is x = 0.
In conclusion, for the absolute value of x to equal negative x, x must be equal to 0.