Solve the following linear inequations in R: 4x+3 2x-5 <6 - Vidyadip

Question

Solve the following linear inequations in R:
$\frac{4\text{x}+3}{2\text{x}-5}<6$

✓

Answer

$\frac{4\text{x}+3}{2\text{x}-5}<6$
$\frac{4\text{x}+3}{2\text{x}-5}-6<0$
$\frac{4\text{x}+3-12\text{x}+30}{2\text{x}-5}<0$
$\frac{-8\text{x}+33}{2\text{x}-5}<0$
$\frac{8\text{x}-33}{2\text{x}-5}>0$
Case: 1 $8\text{x}-33>0$ and $2\text{x}-5>0$
$\Rightarrow\text{x}>\frac{33}{8}$ and $\text{x}>\frac{5}{2}$
$\Rightarrow\text{x}>\frac{33}{8}$
Case 2: $8\text{x}-33<0$ and $2\text{x}-5<0$
$\Rightarrow\text{x}>\frac{33}{8}$ and $\text{x}<\frac{5}{2}$
$\Rightarrow\text{x}<\frac{5}{2}$
Hence the solution set is $\Big(-\infty,\frac{5}{2}\Big)\cup\Big(\frac{33}{8},\infty\Big)$

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