$c)$

$\left|x-\dfrac{5}{6}\right|-\dfrac{1}{3}=\dfrac{1}{4}$

$\left|x-\dfrac{5}{6}\right|=\dfrac{1}{4}+\dfrac{1}{3}$

$\left|x-\dfrac{5}{6}\right|=\dfrac{7}{12}$

Trường hợp 1:

$x-\dfrac{5}{6}=\dfrac{7}{12}$

$x=\dfrac{7}{12}+\dfrac{5}{6}$

$x=\dfrac{17}{12}$

Trường hợp 2:

$x-\dfrac{5}{6}=-\dfrac{7}{12}$

$x=-\dfrac{7}{12}+\dfrac{5}{6}$

$x=\dfrac{1}{4}$

Vậy $x=\dfrac{17}{12}$ hoặc $x=\dfrac{1}{4}$

$d)$

$\dfrac{2}{3}-\left|\dfrac{1}{2}x-1\right|=\dfrac{1}{4}$

$\left|\dfrac{1}{2}x-1\right|=\dfrac{2}{3}-\dfrac{1}{4}$

$\left|\dfrac{1}{2}x-1\right|=\dfrac{5}{12}$

Trường hợp 1:

$\dfrac{1}{2}x-1=\dfrac{5}{12}$

$\dfrac{1}{2}x=\dfrac{5}{12}+1$

$\dfrac{1}{2}x=\dfrac{17}{12}$

$x=\dfrac{17}{12}:\dfrac{1}{2}$

$x=\dfrac{17}{6}$

Trường hợp 2:

$\dfrac{1}{2}x-1=-\dfrac{5}{12}$

$\dfrac{1}{2}x=-\dfrac{5}{12}+1$

$\dfrac{1}{2}x=\dfrac{7}{12}$

$x=\dfrac{7}{12}:\dfrac{1}{2}$

$x=\dfrac{7}{6}$

Vậy $x=\dfrac{17}{6}$ hoặc $x=\dfrac{7}{6}$

$e)$

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

$3x+\dfrac{1}{4}=\left(\dfrac{5}{2}\right)^2$

$3x+\dfrac{1}{4}=\dfrac{25}{4}$

$3x=\dfrac{25}{4}-\dfrac{1}{4}$

$3x=6$

$x=2$

Vậy $x=2$

$f)$

$\left(x-\dfrac{2}{5}\right)\left(3x+\dfrac{1}{2}\right)=0$

Trường hợp 1:

$x-\dfrac{2}{5}=0$

$x=\dfrac{2}{5}$

Trường hợp 2:

$3x+\dfrac{1}{2}=0$

$3x=-\dfrac{1}{2}$

$x=-\dfrac{1}{2}:3$

$x=-\dfrac{1}{6}$

Vậy $x=\dfrac{2}{5},x=-\dfrac{1}{6}$

$g)$

$\left|2x-1\right|=\left|x+\dfrac{1}{2}\right|$

Trường hợp 1:

$2x-1=x+\dfrac{1}{2}$

$2x-x=\dfrac{1}{2}+1$

$x=\dfrac{3}{2}$

Trường hợp 2:

$2x-1=-\left(x+\dfrac{1}{2}\right)$

$2x-1=-x-\dfrac{1}{2}$

$2x+x=-\dfrac{1}{2}+1$

$3x=\dfrac{1}{2}$

$x=\dfrac{1}{2}:3$

$x=\dfrac{1}{6}$

Vậy $x=\dfrac{3}{2},x=\dfrac{1}{6}$

Bài 11:

$a)$

$\left(\dfrac{1}{2}\right)^n=\dfrac{1}{32}$

$\left(\dfrac{1}{2}\right)^n=\left(\dfrac{1}{2}\right)^5$

$n=5$

Vậy $n=5$

$b)$

$\dfrac{1}{3}\cdot9^n=3^7$

$9^n=3^7:\dfrac{1}{3}$

$9^n=3^7\cdot3$

$9^n=3^8$

$(3^2)^n=3^8$

$3^{2n}=3^8$

$2n=8$

$n=4$

Vậy $n=4$

$c)$

$\dfrac{81}{3^n}=3$

$3^n=81:3$

$3^n=27$

$3^n=3^3$

$n=3$

Vậy $n=3$

$d)$

$2^{n+3}+2^n=144$

$2^n\cdot2^3+2^n=144$

$2^n\cdot(2^3+1)=144$

$2^n\cdot(8+1)=144$

$2^n\cdot9=144$

$2^n=144:9$

$2^n=16$

$2^n=2^4$

$n=4$

Vậy $n=4$

$e)$

$\left(n-\dfrac{3}{2}\right)^3=-\dfrac{27}{8}$

$\left(n-\dfrac{3}{2}\right)^3=\left(-\dfrac{3}{2}\right)^3$

$n-\dfrac{3}{2}=-\dfrac{3}{2}$

$n=-\dfrac{3}{2}+\dfrac{3}{2}$

$n=0$

Vậy $n=0$

$f)$

$27<3^n\le243$

$3^3<3^n\le3^5$

$3<n\le5$

Vì $n$ là số nguyên nên $n\in\{4;5\}$

Vậy $n\in\{4;5\}$