The number of integral solutions x of _x+72(x-72x-3)^20 is:

Question

The number of integral solutions $x$ of $\log_{x+\frac72}\left(\dfrac{x-7}{2x-3}\right)^2\ge0$ is:

Accepted Answer

The base requires $x+\dfrac72>0,\ x eq-\dfrac52$, i.e. $x\in\left(-\dfrac72,\infty ight)\setminus\left\{-\dfrac52 ight\}$. Case base $>1$ ($x>-\dfrac52$): need $\left(\dfrac{x-7}{2x-3} ight)^2\ge1 \implies (x-7)^2\ge(2x-3)^2 \implies (3x-10)(x+4)\le0 \implies x\in\left[-4,\dfrac{10}3 ight]$. Intersected with $x>-\dfrac52$: $x\in\left(-\dfrac52,\dfrac{10}3 ight]$. Case $0<$ base $<1$ ($-\dfrac72

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