Question

I need help with this practice problem solving The subject is trig

Answer 1

Step-by-step explanation

Given

`\begin{gathered} |P|=(3)/(4)x+2 \\ |Q|=2x+1 \\ |P+Q|=(2)/(3)x-2 \end{gathered}`

We know that, from Cauchy inequalities

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

Therefore;

`\begin{gathered} (3)/(4)x+2+2x+1>(2)/(3)x-2 \\ (3)/(4)x+2x+3>(2)/(3)x-2 \\ \text{Multiply through by 12} \\ 12((3)/(4)x+2x+3)>12((2)/(3)x-2) \\ 9x+24x+36>8x-24 \\ 33x+36>8x-24 \\ 33x-8x>-36-24 \\ 25x>-60 \\ x>-(12)/(5) \\ \end{gathered}`

From the above, we will find the x values for the given inequalities

`\begin{gathered} \text{For }(3)/(4)x+2>0 \\ (3)/(4)x>-2 \\ x>-(8)/(3) \end{gathered}`

`\begin{gathered} \text{For }2x+1 \\ 2x>-1 \\ x>-(1)/(2) \end{gathered}`
`\begin{gathered} \text{For }(2)/(3)x-2>0 \\ (2)/(3)x>2 \\ x>3 \end{gathered}`

The range that satisfies the above inequality is

Answer:

`(3,0)`