for each mathematical statement below determine if it's always, sometimes, or never true.
a. |m-n| = |m| - |n|
b. b + (-b) = 0
|6-4| = |6|-|4| = 6-4 = 2
|4-6| ≠ |4|-|6| = 4-6 = -2
-b is the additive inverse of b, so the statement is always true.
thanks
To determine if a mathematical statement is always, sometimes, or never true, we have to analyze the statement and consider all possible values of the variables involved. Let's analyze each statement:
a. |m - n| = |m| - |n|
To determine if this statement is always, sometimes, or never true, we need to consider all possible values for m and n.
If m and n are both positive numbers, the equation becomes |m - n| = m - n, |m| - |n|. In this case, the statement is sometimes true because |m - n| is equal to |m| - |n|.
If m is positive and n is a negative number, the equation becomes |m - n| = m + n, |m| - (-n). In this case, the statement is sometimes true since |m - n| is equal to m + n, |m| - (-n).
If m is a negative number and n is positive, the equation becomes |m - n| = -(m - n), -|m| - |n|. In this case, the statement is sometimes true since |m - n| is equal to -(m - n), -|m| - |n|.
If both m and n are negative numbers, the equation becomes |m - n| = -(m - n), -|m| - (-n). In this case, the statement is sometimes true since |m - n| is equal to -(m - n), -|m| - (-n).
So, the statement |m - n| = |m| - |n| is sometimes true, depending on the values of m and n.
b. b + (-b) = 0
To determine if this statement is always, sometimes, or never true, we need to consider all possible values for b.
In every case, b + (-b) is equal to 0. So, the statement b + (-b) = 0 is always true for any value of b.
Thus, the statement b + (-b) = 0 is always true.