Question

The range of values of $x$ which satisfy the inequation ${\log }_5(x^2-11x+43)<2$ is

• A. $x\in (2,9)$
• B. $x\in [2,9]$
• C. $x\in (-\infty ,2)\cup (9,\infty )$
• D. $x\in (1,9)$

Answer 1

Answer: (A)

$x\in (2,9)$

For the inequality
${\log }_5(x^2-11x+43)<2$,

for $\log$ to be defined
$x^2-11x+43>0$ ....(1)

and

${\log }_5(x^2-11x+43)<2$,

$x^2-11x+43<5^2$

$\Rightarrow x^2-11x+43<25$

$\Rightarrow x^2-11x+18<0$ ...(2)

The inequality (1) in equivalent to

$x^2-2\cdot \frac{11}{2}\cdot x+(\frac{11}{2}{)}^2-(\frac{11}{2}{)}^2+43>0$

${(x-\frac{11}{2})}^2+\frac{51}{4}>0$ for all $x\in R\Rightarrow x\in (-\infty ,\infty )$ ....(3)

The inequality (2) is

$\Rightarrow x^2-9x-2x+18<0$
$(x-2)(x-9)<0$

i.e., $2<x<9$ ...(4)

From (3) and (4), the required solution is
$2<x<9$

i.e., $x\in (2,9)$