Solve -3 x -2- -1-2- x - R

We have, |x|≤ a⇔ a≤ x ≤ a. ∴ |3x 2| ≤1/2, ⇔ 1/2≤ 3x 2≤1/2 ⇔ 1/2≤ 3x 2 and 3x 2≤1/2 ⇔ 1/2+2≤ 3x and 3x≤1/2+2 ⇔3/2≤ 3x and 3x≤5/2 ⇔1/2≤ x and x≤5/6 1/2≤ x≤5/6 ∴ ...

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Byju's Answer

Standard XI

Mathematics

Properties of Inequalities

Solve |3 x -2...

Question

Solve $| 3 x - 2 | \leq \frac{1}{2}, x ϵ R$

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Solution

We have,

$| x | \leq a \Leftrightarrow - a \leq x \leq a$ .

$∴ | 3 x - 2 | \leq \frac{1}{2}, \Leftrightarrow - \frac{1}{2} \leq 3 x - 2 \leq \frac{1}{2}$

$\Leftrightarrow \frac{- 1}{2} \leq 3 x - 2 a n d 3 x - 2 \leq \frac{1}{2}$

$\Leftrightarrow \frac{- 1}{2} + 2 \leq 3 x a n d 3 x \leq \frac{1}{2} + 2$

$\Leftrightarrow \frac{3}{2} \leq 3 x a n d 3 x \leq \frac{5}{2}$

$\Leftrightarrow \frac{1}{2} \leq x a n d x \leq \frac{5}{6}$

$\frac{1}{2} \leq x \leq \frac{5}{6}$

$∴$ solution set $= {x ϵ R : \frac{1}{2} \leq x \leq \frac{5}{6}} = [\frac{1}{2}, \frac{5}{6}]$

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Solve |3x−2|≤12,xϵR

We have, |x|≤a⇔−a≤x≤a. ∴ |3x−2|≤12,⇔−12≤3x−2≤12 ⇔−12≤3x−2 and 3x−2≤12 ⇔−12+2≤3x and 3x≤12+2 ⇔32≤3x and 3x≤52 ⇔12≤x and x≤56 12≤x≤56 ∴ solution set ={xϵR:12≤x≤56}=[12,56]

Solve $| 3 x - 2 | \leq \frac{1}{2}, x ϵ R$