Answer:
[tex]\boxed{\boxed{\bf y < - \cfrac{1}{2}}} [/tex]
Option A
Step-by-step explanation:
[tex] \bf \: Given \: inequality :[/tex]
[tex]\sf \implies \: \cfrac{4}{3 } \: y < \cfrac{ - 8}{3} \: y[/tex]
We need to find the solution to the inequality.
[tex] \bf \: Solution:[/tex]
[tex]\sf \implies \: \cfrac{4}{3 } \: y < \cfrac{ - 8}{3} \: y[/tex]
[tex] \rm \: Firstly,Flip \: the \: inequality :[/tex]
[tex]\sf \implies \cfrac{ - 8}{3} y > \cfrac{4}{3} [/tex]
[tex] \rm \: Then,\; multiply\; each\: side \: \: by \: \cfrac{ - 3}{8} \: : [/tex]
[tex]\sf \implies \: \cfrac{ - 8}{3}y \times \cfrac{ 3}{ - 8} > \cfrac{4}{3} \times \cfrac{ 3}{ - 8} [/tex]
[tex] \rm \: Use \: cancellation \: method \: to \: cancel \: LHS:-[/tex]
Steps of cancelling :-
- Cancel -8( which is on the numerator) and -8 (on the denominator) :
[tex]\sf \implies \cfrac{ \cancel{- 8}}{3}y \times \cfrac{ 3}{ \cancel{- 8} } > \cfrac{4}{3} \times \cfrac{ 3}{ - 8} [/tex]
- Cancel 3(which is on the numerator) and 3( which is on the denominator) :
[tex]\sf \implies\cfrac{ \cancel{- 8}}{ \cancel3}y \times \cfrac{ \cancel 3}{ \cancel{- 8} } > \cfrac{4}{3} \times \cfrac{ 3}{ - 8} [/tex]
[tex] \sf \implies \: 1y < \cfrac{4}{3} \times \cfrac{3}{ - 8} [/tex]
As we know 1y equals to y. So,
[tex] \sf \implies \: y < \cfrac{4}{3} \times \cfrac{3}{ - 8} [/tex]
[tex] \rm \: Now, Cancel \: the \: RHS :[/tex]
Steps of cancelling:-
- Cancel 3 (which is on the numerator) and cancel 3 (which is on the denominator):
[tex] \sf \implies \: y < \cfrac{4}{ \cancel3} \times \cfrac{ \cancel3}{ - 8} [/tex]
[tex] \sf \implies{y} < 4 \times \cfrac{1}{ - 8} [/tex]
[tex] \sf \implies \: y < \cancel{4} \times \cfrac{1}{ \cancel{ - 8}} [/tex]
[tex] \sf \implies \: y < 1 × \cfrac{ - 1}{2} [/tex]
[tex] \sf \implies \: y < \cfrac{ - 1}{2} [/tex]
[tex] \rm \: Which \: can \: be \: rewritten \: as,[/tex]
[tex] \sf \implies \: y < - \cfrac{1}{2} [/tex]
This matches with option A.
Hence, Option A is correct!
[tex] \rule{225pt}{2pt}[/tex]
I hope this helps!
Let me know if you have any questions.I am joyous to help!
