4x+32x 5 lt;6⇒4x+32x 5 6 lt;0⇒4x+3 62x 52x 5 lt;0⇒4x+3 12x+302x 5 lt;0⇒ 8x+332x 5 lt;0⇒8x 332x 5 gt;0 x∈ ∞,52∪338,∞ ...

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Differentiability

4 x+32 x-5<6

Question

$\frac{4 x + 3}{2 x - 5} < 6$

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Solution

$\frac{4 x + 3}{2 x - 5} < 6 \Rightarrow \frac{4 x + 3}{2 x - 5} - 6 < 0 \Rightarrow \frac{4 x + 3 - 6 (2 x - 5)}{2 x - 5} < 0 \Rightarrow \frac{4 x + 3 - 12 x + 30}{2 x - 5} < 0 \Rightarrow \frac{- 8 x + 33}{2 x - 5} < 0 \Rightarrow \frac{8 x - 33}{2 x - 5} > 0$

$x \in (- \infty, \frac{5}{2}) \cup (\frac{33}{8}, \infty)$

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