$2$ times $x$ minus $3$ divided by $4$ times $x$ plus $5$ equals to $1$ by $3$ is a given equation in algebraic form. The value of $x$ is unknown and it should be evaluated by solving this equation in this problem.

$\dfrac{2x-3}{4x+5}$ $\,=\,$ $\dfrac{1}{3}$

The given equation in rational form is already in simplified form. $2x-3$ divided by $4x+5$ is a rational expression on one side and $1$ divided by $3$ is a fraction on other side of the given equation. It can be simplified by using the cross multiply method.

$\implies$ $3 \times (2x-3)$ $\,=\,$ $1 \times (4x+5)$

On the left-hand side of the equation, the coefficient $3$ can be distributed over the difference of the terms $2x$ and $3$ as per the distributive property of multiplication over the subtraction. On the right-hand side of the equation, the coefficient $1$ can be distributed across the addition of the terms $4x$ and $5$ as per the distributive property of multiplication over the addition.

$\implies$ $3 \times 2x$ $-$ $3 \times 3$ $\,=\,$ $1 \times 4x$ $+$ $1 \times 5$

$\implies$ $6x-9$ $\,=\,$ $4x+5$

The equation $6$ times $x$ minus $9$ equals to $4$ times $x$ plus $5$ is a linear equation in one variable. It can be solved by using the transposition method.

$\implies$ $6x-4x$ $\,=\,$ $5+9$

$\implies$ $2x$ $\,=\,$ $14$

$\implies$ $2 \times x$ $\,=\,$ $14$

$\implies$ $x$ $\,=\,$ $\dfrac{14}{2}$

$\implies$ $x$ $\,=\,$ $\dfrac{\cancel{14}}{\cancel{2}}$

$\,\,\,\therefore\,\,\,\,\,\,$ $x\,=\,7$

Therefore, the solution set for the variable $x\,=\,\{\,7\,\}$

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