Question

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

On what interval must x fall?

[−5/2,0]
[−1,∞)
(−∞,−5/2]
[−1,0]

Answer 1

Answer:

Step-by-step explanation:

Given that,

|P| = 2x / 5 + 1

|Q| = x + 1

|P+Q| = 3x + 2

We know that, from Cauchy inequalities

|P| + |Q| > |P+Q|

(2x / 5) + 1 + x + 1 > 3x + 2

(2x / 5) + 2 + x > 3x + 2

Multiply through by 5

2x + 10 + 5x > 15x + 10

Collect like terms

2x + 5x - 15x > 10 -10

-8x > 0

Then,

Divide both side by -8, since we are dividing with a negative number the inequality sign will change

Then,

x < 0 first condition

Note that,

|P| > 0

Then,

2x / 5 + 1 > 0

2x / 5 > -1

2x > -5

x > -5/2

Also,

|Q| > 0

x + 1 > 0

x > -1

Also

|P+Q| > 0

3x + 2 > 0

3x > -2

x > -2 / 3

So, comparing the four conditions

We see that x ranges from -5/2 to 0, this range covers all the other ranges

(-5/2, 0)

The first answer is correct