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?

a) [−4/3, −60/47]

b) [-2, −4/3]

c) [−2, [infinity])

d) (−[infinity], −4/3)


Please Help!!

Answer 1

Answer: (b)

Explanation:

Given

`\left | p\right |=(2)/(5)x+2`

`\left | q\right |=(-3)/(4)x-1`

`\left | p+q\right |=2x+4`

for two complex number,
`z_1,z_2`

`\left | z_1+z_2\right |\leq \left | z_1\right |+\left | z_2\right |`

Apply the above the property

`\Rightarrow 2x+4\leq (2)/(5)x+2-(3)/(4)x-1\\\\\Rightarrow 2x+4\leq (8x-15x)/(20)+1\\\\\Rightarrow 2x+3\leq -(7x)/(20)\\\Rightarrow x\leq -(60)/(47)`

Also, the absolute value of each complex number must be greater than or equal to zero

`\text{Case-1}\\\\\Rightarrow (2)/(5)x+2\geq 0\\\\\Rightarrow x\geq -5\\\\\text{Case-2}\\\\\Rightarrow -(3)/(4)x-1\geq 0\\\\\Rightarrow x\leq -(4)/(3)\\\\\text{Case-3}\\\\\Rightarrow 2x+4\geq 0\\\Rightarrow x\geq -2`

Taking the intersection of the above values of x, we get

`x\in [-2,-(4)/(3)]`