Question

If $|4-3x|\le \frac{1}{2}$ then $x$ is equal to

• A. $[\frac{7}{6},\frac{3}{2})$
• B. $(\frac{7}{6},\frac{3}{2})$
• C. $[\frac{7}{6},\frac{3}{2}]$
• D. $None$ $of$ $these$

Answer 1

The correct option is C $[\frac{7}{6},\frac{3}{2}]$

Given $|4-3x|\le \frac{1}{2}$

$\Rightarrow |3x-4|\le \frac{1}{2}$ [using $|x-y|=|y-x|$]

Using property $|x|\le a\Rightarrow -a\le x\le a$

$\Rightarrow -\frac{1}{2}\le 3x-4\le \frac{1}{2}$

[adding $4$ to the equation]

$\Rightarrow -\frac{1}{2}+4\le 3x-4+4\le \frac{1}{2}+4$

$\Rightarrow \frac{7}{2}\le 3x\le \frac{9}{2}$

[Dividing by $3$]

$\Rightarrow \frac{7}{2}\times \frac{1}{3}\le \frac{3x}{3}\le \frac{9}{2}\times \frac{1}{3}$

$\Rightarrow \frac{7}{6}\le x\le \frac{3}{2}$

$\Rightarrow x\in [\frac{7}{6},\frac{3}{2}]$.