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?a. [−2/3, 0]b. [−5/2,0]c. (−∞,−5/2]d. [−2/3, ∞)

Answer 1

`[-(2)/(3),0]`

Step-by-step explanation

Given:

`\begin{gathered} |p|=(2)/(5)x+1 \\ |q|=x+1 \\ |p+q|=3x+2 \end{gathered}`

Desired Outcome:

x-intervals

Applying Cauchy Inequalities for condition 1

`\begin{gathered} |p|+|q|>|p+q| \\ (2x)/(5)+1+x+1>3x+2 \\ \text{ Find the LCM} \\ (2x+5+5x+5)/(5)>3x+2 \\ (7x+10)/(5)>3x+2 \\ \text{ Cross-multiply} \\ 7x+10>5(3x+2) \\ 7x+10>15x+10 \\ 7x-15x>10-10 \\ -8x>0 \\ \text{ Divide through by -8} \\ x<0 \end{gathered}`

For Condition 2

`\begin{gathered} |p|>0 \\ (2)/(5)x+1>0 \\ \text{ Find LCM} \\ (2x+5)/(5)>0 \\ 2x+5>0 \\ 2x>-5 \\ x>-(5)/(2) \end{gathered}`

For Condition 3

`\begin{gathered} |p+q|>0 \\ 3x+2>0 \\ 3x>-2 \\ x>-(2)/(3) \end{gathered}`

For Condition 4