- #1
Steve Turchin
- 11
- 0
Solve the following inequality. Represent your answer graphically:
## |z-1| + |z-5| < 4 ##
## z = a + bi \\
|x+y| \leq |x| + |y| ## Triangle inequality
## |z-1| + |z-5| < 4 \\
\\
x = z-1 \ \ , \ \ y = z-5 \\
\\
|z-1+z-5| \leq |z-1| + |z-5| \\
|2z-6| \leq |z-1| + |z-5| \lt 4 \ \ \Leftrightarrow \ \ |2z-6| \lt 4 \\
2z-6 \lt 4 \ \ , \ \ -(2z-6) \lt 4 \\
z \lt 5 \ \ , \ \ \ \ \ \ \ \ \ \ \ \ -2z + 6 \lt 4 \ \ \ \Leftrightarrow \ \ \ -z \lt -2 \ \ \ \Leftrightarrow \ \ \ z \gt 1 \\
z \lt 5 \ \cap \ z \gt 1 \ \ \ \Leftrightarrow \ \ \ 1 \lt z \lt 5 \\
1 \lt a+bi \lt 5 \ \ \Rightarrow \ \ \ 1 \lt a+bi \lt 5 \\
1 \lt a \lt 5 \ \ \ \ \ , \ \ \ \ \ \ \ b = 0 \ \ \ \ for \ \ a,b \in \mathfrak R
##
I think this is basically the interval ## (1,5) ## on the Real axis.
I got this far, I doubt this is correct. Any tip on what a graphical representation of this would be?
Thanks in advance.
## |z-1| + |z-5| < 4 ##
Homework Equations
## z = a + bi \\
|x+y| \leq |x| + |y| ## Triangle inequality
The Attempt at a Solution
## |z-1| + |z-5| < 4 \\
\\
x = z-1 \ \ , \ \ y = z-5 \\
\\
|z-1+z-5| \leq |z-1| + |z-5| \\
|2z-6| \leq |z-1| + |z-5| \lt 4 \ \ \Leftrightarrow \ \ |2z-6| \lt 4 \\
2z-6 \lt 4 \ \ , \ \ -(2z-6) \lt 4 \\
z \lt 5 \ \ , \ \ \ \ \ \ \ \ \ \ \ \ -2z + 6 \lt 4 \ \ \ \Leftrightarrow \ \ \ -z \lt -2 \ \ \ \Leftrightarrow \ \ \ z \gt 1 \\
z \lt 5 \ \cap \ z \gt 1 \ \ \ \Leftrightarrow \ \ \ 1 \lt z \lt 5 \\
1 \lt a+bi \lt 5 \ \ \Rightarrow \ \ \ 1 \lt a+bi \lt 5 \\
1 \lt a \lt 5 \ \ \ \ \ , \ \ \ \ \ \ \ b = 0 \ \ \ \ for \ \ a,b \in \mathfrak R
##
I think this is basically the interval ## (1,5) ## on the Real axis.
I got this far, I doubt this is correct. Any tip on what a graphical representation of this would be?
Thanks in advance.