Question

The solution set of the equation $\left[2x\right]=\left[x\right]+3,0\le x\le 4$ where $\left[x\right]=$ the greatest integer less than or equal to $x$, is

• A. $\left\{3\right\}$
• B. $\{3,\frac{5}{2}\}$
• C. $[\frac{5}{2},\frac{7}{2})$
• D. $[2,3]$

Answer 1

The correct option is C $[\frac{5}{2},\frac{7}{2})$

Let $\left\{x\right\}$ be the fraction part of $x$.

If $0\le \left\{x\right\}<\frac{1}{2}$, then $\left[2x\right]=2\left[x\right]$.

Hence the given equation can be written as

$2\left[x\right]=\left[x\right]+3$

$\therefore \left[x\right]=3$. Hence the solution set for the equation is

$\left[3,\frac{7}{2}\right)$ as the fraction part of $x$ is less than half.

If $\frac{1}{2}\le \left\{x\right\}<1$, then $\left[2x\right]=2\left[x\right]+1$.

Hence the given equation can be written as

$2\left[x\right]+1=\left[x\right]+3$

$\therefore \left[x\right]=2$. Hence the solution set for the equation is

$\left[\frac{5}{2},3\right)$ as the fraction part of $x$ is greater than half.

Hence, the solution set of the equation is $\left[\frac{5}{3},\frac{7}{2}\right)$