Question

Please answer the question correctly ​

Answer 1

Given: 3x / 5 + 2 < x + 4 <= x / 2 + 5 , x in R

Given: 3x / 5 + 2 < x + 4 <= x / 2 + 5 , x in RTo find We have, 3x / 5 + 2 < x + 4 <= x / 2 + 5 , x notin R

Case 1:

=>3x/5+2<x+4 ,x € R =>x+4>3x/5+2 , x € R =>x-3x/5>2-4 , x € R Rightarrow(5x-3x)/5>-2 , x in R Rightarrow2*/5>-2 , x in R Rightarrow x>-2*(5/2) , x in R =>x>-5 , x in R Rightarrow* in(-5, infty) Solution set..

Case 2:

=>x+4<= x/2+5 , x in R =>x-x/2<=5-4 ,x € R Rightarrow(2x-x)/2<=1 , x notin R Rightarrow>x/2<=1 , x in R =>x<=1.2 , x in R Rightarrow>x<=2 , x in R Rightarrow x in( infty,2] → Here, the intersection set of case1 and case2 will give the solution set. Thus, Solution set will be given as ; Rightarrow* in(-5, infty) and x in( infty,2] * in(-5, infty) cap( infty,2] → * in(-5,2] (ie . -5<x<=2,x in R)

Hence, Solution set : x notin(-5,2] (ie . -5<x<=2,x in R) .

Answer 2

$\frac{3x}{5} + 2 < x + 4 ≤ \frac{x}{2} + 5$ x € R

$\frac{3x}{5} + 2 < x + 4$

=> $\frac{3x+10}{5} < x + 4$

=> $3x + 10 < 5x + 20$

=> $10 - 20 < 5x - 3x$

=> $- 10 < 2x$

=> $\frac{-10}{2} < x$

=> $-5 < x$

$x + 4 ≤ \frac{x}{2} + 5$

=> $x + 4 ≤ \frac{x + 10}{2}$

=> $2x + 8 ≤ x + 10$

=> $2x - x ≤ 10 - 8$

=> $x ≤ 2$

S.S. => { x : -5 < x ≤ 2 , x € R}

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