Question

Solve the inequality
${}^{x-1}{C}_4-{}^{x-1}{C}_3-\frac{5}{4}{}^{x-2}{C}_2<0,x\in N$. Find difference of largest and smallest values x can take ?

Answer 1

`We have
${}^{x-1}{C}_4-{}^{x-1}{C}_3-\frac{5}{4}{}^{x-2}{C}_2<0$
$\iff \frac{(x-1)(x-2)(x-3)(x-4)}{\mathrm{1.2.3.4}}-\frac{(x-1)(x-2)(x-3)}{\mathrm{1.2.3}}-\frac{5}{4}.(x-2)(x-3)<0$
$\iff (x-1)(x-2)(x-3)(x-4)-4(x-1)(x-2)(x-3)-30(x-2)(x-3)<0$
$\iff (x-2)(x-3)\left\{\right(x-1\left)\right(x-4)-4(x-1)-30\}<0$
$\iff (x-2)(x-3)\{x^2-9x-22\}<0$
$\iff (x-2)(x-3)(x+2)(x-11)<0$
From wavy curve method
$x\in (-2,2)\cup (3,11)$
But $x\in N$
$\therefore x=1,4,5,6,7,8,9,10...\left(1\right)$
From inequality,
$x-1\ge 4,x-1\ge 3,x-2\ge 2$
or $x\ge 5,x\ge 4,x\ge 4$
Hence $x\ge 5...\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$, solutions of the inequality
$x=5,6,7,8,9,10$
$\therefore$ the difference between the largest and smallest value of $x$ is $5$