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Question
Solve the following inequation and represent the solution set on the number line : `-3 < -(1)/(2) - (2x)/(3) ≤ (5)/(6), x ∈ "R"`
Solution
`-3 < -(1)/(2) - (2x)/(3) ≤ (5)/(6), x ∈ "R"`
(i) `-3 < -(1)/(2) - (2x)/(3)`
⇒ `-3 ≤ - (1/2 + (2x)/3)`
⇒ `- (1/2 + (2x)/3) > -3`
⇒ `-(2x)/(3) > -3 + (1)/(2)`
⇒ `-(2x)/(3) > (-5)/(2)`
⇒ `(2x)/(3) < (5)/(2)`
⇒ `x < (5)/(2) xx (3)/(2)`
⇒ `x < (15)/(4)` ....(i)
(ii) `-(1)/(2) - (2x)/(3) ≤ (5)/(6)`
⇒ `-(2x)/(3) ≤ (5)/(6) + (1)/(2)`
⇒ `(-2x)/(3) ≤ (5 + 3)/(6)`
⇒ `(-2)/(3) xx ≤ (8)/(6)`
⇒ `(2)/(3)x ≥ (-8)/(6)`
⇒ `x ≥ (-8)/(6) xx (3)/(2)`
⇒ x ≥ -2
⇒ -2 ≤ x ....(ii)
⇒ From (i) and (ii),
`-2 ≤ ≤ (15)/(4)`
∴ Solution = `{ x : x ∈ "R", -2 ≤ x < (15)/(4)}`
Now solution on number line
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