Question

Please answer.NO SPAM​

Answer 1

Step-by-step explanation:

$\frac{m {x}^{2} + 3x + 4}{ {x}^{2} + 2x + 2} < 5$

$\implies \: \frac{m {x}^{2} + 3x + 4}{ {x}^{2} + 2x + 2} - 5 < 0$

$\implies \: \frac{(m {x}^{2} + 3x + 4) - 5( {x}^{2} + 2x + 2) }{ {x}^{2} + 2x + 2} < 0$

$\implies \: \frac{(m - 5){x}^{2} - 7x - 6}{ {x}^{2} + 2x + 2} < 0$

Now, in the denominator,

the quadratic equation x²+2x+2 is always positive, as its a>0 and D<0

So, numerator's quadratic equation must be negative,

$m - 5 < 0 \: \: and \: \: ( - 7) ^{2} - 4.(m - 5).( - 6) < 0$

$\implies \: m < 5 \: \: and \: \: 49 + 24(m - 5) < 0$

$\implies \: m < 5 \: \: and \: \: 49 + 24m - 120 < 0$

$\implies \: m < 5 \: \: and \: \: 24m - 71< 0$

$\implies \: m < 5 \: \: and \: \: m < \frac{71}{24}$

Hence, $m < \frac{71}{24}$