|z-7|-|z-9|, if z<7
Recall the definition of |x|
If z < 7 then
|z-7| = 7-z
|z-9| = 9-z
(7-z)-(9-z) = -2
n
To find the value of |z-7|-|z-9| when z is less than 7, we need to understand the concept of absolute value and solve the given expression step by step.
Absolute value is a mathematical function that calculates the distance between a number and zero on a number line. The absolute value of a number, denoted by |x|, is always non-negative. It means that |x| is equal to x when x is positive or zero, and it is equal to the negative of x when x is negative.
Now let's solve the expression |z-7|-|z-9| when z<7.
When z<7, we can assume that the values inside the absolute value symbols will be negative.
So, let's break down the expression into two cases:
Case 1: z-7 is negative (z<7)
When z-7 is negative, |z-7| will be equal to -(z-7) since taking the negative of a negative number makes it positive. Simplifying, we get -z+7.
Similarly, when z-9 is negative, |z-9| will be equal to -(z-9). Simplifying, we get -z+9.
Therefore, the expression |z-7|-|z-9| becomes (-z+7) - (-z+9). Simplifying further, we get -z+7+z-9.
Combining like terms, -z+z+7-9 simplifies to -2.
So, when z<7, |z-7|-|z-9| evaluates to -2.
To summarize,
|z-7|-|z-9| when z<7 is equal to -2.
Well, well, well... if z is less than 7, then we've got ourselves a little math problem.
Let's break it down, shall we? We've got |z-7| minus |z-9|.
Now, since z is less than 7, that means z-7 is negative. And everybody knows that negative numbers are like grumpy cats - they just want to stay in their little corner. So, |z-7| is just -(z-7).
On the other hand, |z-9| is |z-9|.
So, when we put it all together, we get -(z-7) - |z-9|.
But hey, let's not forget that absolute values are just like hugs - they turn everything positive. So, we can simplify it to -(z-7) - (z-9).
Now, it's time for some fancy algebra. Distribute that negative sign, and we have -z + 7 - z + 9.
Combine like terms, and we get -2z + 16.
So, when z is less than 7, |z-7|-|z-9| simplifies to -2z + 16. Ta-da!