Answer to Question #192952 in Algebra for oxman

Question

3/4𝑥+3≤2/3≤2/𝑥+5

Answer 1

${3\over 4x} +3\leq {2\over 3}\leq {2\over x}+5$

Split into two and solve separately

1). ${3\over 4x}+3\leq {2\over 3}$

Multiply through by 4x

${3\over 4x}(4x)+3(4x)\leq {2(4x)\over 3}$

$\implies 3+12x\leq {8x\over 3}$

Multiply through by 3

$3(3)+12x(3)\leq{8x\over 3}(3)$

$\implies 9+36x\leq8x$

$9-9+36x-8x\leq8x-8x-9$

$\implies 28x\leq -9\implies x\leq {-9\over 28}$

2). ${2\over 3}\leq{2\over x}+5$

Multiply through by 3x

${2\over 3}(3x)\leq{2\over x}(3x)+5(3x)$

$\implies 2x\leq 6+15x \implies 2x-2x-6\leq6-6+15x-24$

$\implies -6\leq 13 x \implies x\geq {-6\over 13}$

Combining the two,we have

${-6\over 13}\leq x\leq {-9\over 28}$