Question

3x/5 + 2 < x+4 <= x/2 + 5 ,x € R
solve the following inequation and represent the soulution set on number line ​

Answer 1

Required answer

$-10<x\leq 2$

To solve

Inequality $\dfrac{3x}{5} <x+4\leq \dfrac{x}{2} +5$

Before we solve

$A<B<C$ is equivalent to $A<B$ and $B<C$. In other words, we can solve two inequality.

Solution

$\dfrac{3x}{5} <x+4$ ...[I]

$x+4\leq \dfrac{x}{2} +5$ ...[II]

Solving [I]

$\rightarrow \dfrac{3x}{5} <x+4$

$\rightarrow 3x <5(x+4)$

$\rightarrow 3x <5x+20$

$\rightarrow -2x<20$

$\rightarrow \boxed{x>-10}$

Solving [II]

$\rightarrow x+4\leq \dfrac{x}{2} +5$

$\rightarrow 2(x+4)\leq 2(\dfrac{x}{2} +5)$

$\rightarrow 2x+8\leq x+10$

$\rightarrow \boxed{x\leq 2}$

Answer 2

$\huge\bf{{\color{indigo}{A}}{\color{maroon}{ñ}}{\red{s}}{\color{red}{w}}{\color{orange}{ê}}{\color{gold}{Я࿐}}}$

$\dfrac{3x}{5} < x+4 [I] \\ x+4\leq \dfrac{x}{2} +5x+4[II] \\ \sf Solving [I] \\ \rightarrow \dfrac{3x}{5} \\ \rightarrow 3x < 5(x+4) \\ \rightarrow 3x < 5x+20\rightarrow -2x < 20 \\\rightarrow \boxed{x > -10} \\ \sf Solving [II] \\ \rightarrow x+4\leq \dfrac{x}{2} +5 \\ \rightarrow 2(x+4)\leq 2(\dfrac{x}{2} +5) \\ \rightarrow 2x+8\leq x+10 \\ \rightarrow \boxed{x\leq 2}$