Question

p and q are complex numbers such that |p|=2/5 x+2 , |q|=−3/4 x−1 , and |p+q|=2x+4 .

On what interval must x fall?

(−∞,−4/3]

[−4/3,−60/47]

[−2,∞)

[−2,−4/3]

Answer 1

@darkknight

Answer 2

So, You can write four inequalities
2/5x+2 >= 0,
-3/4x-1 >= 0,
2x+4 >= 0,
2x+4 <= (2/5x+2) + (-3/4x-1).
First three are because of the nonnegativity of the modulus, the fourth is due to the triangle inequality. We need to satisfy all four So by simplifying each and taking the intersection, we will get the smallest interval possible containing x.
So simplify the inequalities first

Answer 3

The simplified inequalities are
x >= -5,

x <= -4/3,

x >= -2,

x <= -60/47.
out of these we can drop the first because that would be implied by the third and drop the fourth because that is implied by the second.

Answer 4

So in the end you have
-2 <= x <= -4/3.
Or [-2, -4/3]

Answer 5

Okay boomer, did that help though?

Answer 6

thank you so much for your explanation , it really helps me grasp the material rather than just getting the answer, this makes sense. and I can now see the error I have made. Thank you again! I am gonna take note of this, I dont have any further questions you explained it really well.

Answer 7

You are amazing! I have one more question ,if you could look at it .