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

ANSWER

$[-\frac{2}{3},0]$

EXPLANATION

Given:

$\begin{gathered} |p|=\frac{2}{5}x+1 \\ |q|=x+1 \\ |p+q|=3x+2 \end{gathered}$

Desired Outcome:

x-intervals

Applying Cauchy Inequalities for condition 1

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

For Condition 2

$\begin{gathered} |p|>0 \\ \frac{2}{5}x+1>0 \\ \text{ Find LCM} \\ \frac{2x+5}{5}>0 \\ 2x+5>0 \\ 2x>-5 \\ x>-\frac{5}{2} \end{gathered}$

For Condition 3

$\begin{gathered} |p+q|>0 \\ 3x+2>0 \\ 3x>-2 \\ x>-\frac{2}{3} \end{gathered}$

For Condition 4

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