ĐKXĐ: x<>0

\(\dfrac{2}{5}\left(\dfrac{1}{2}-\dfrac{2}{2x}\right)-\dfrac{3}{5}\left(\dfrac{1}{3x}-\dfrac{10}{3}\right)=-\dfrac{1}{4}\)

=>\(\dfrac{1}{5}-\dfrac{2}{5x}-\dfrac{1}{5x}+2=-\dfrac{1}{4}\)

=>\(-\dfrac{3}{5x}+\dfrac{11}{5}=-\dfrac{1}{4}\)

=>\(-\dfrac{3}{5x}=-\dfrac{1}{4}-\dfrac{11}{5}=\dfrac{-49}{20}\)

=>\(\dfrac{3}{5x}=\dfrac{49}{20}\)

=>\(5x=20\cdot\dfrac{3}{49}=\dfrac{60}{49}\)

=>\(x=\dfrac{12}{49}\left(nhận\right)\)

\(A=\dfrac{1.2+2.1.2.2+3.1.3.2+4.1.4.2+5.1.5.2}{3.4+2.3.2.4+3.3.3.4+4.3.4.4+5.3.5.4}\)

\(=\dfrac{1.2+1.2.2^2+1.2.3^2+1.2.4^2+1.2.5^2}{3.4+3.4.2^2+3.4.3^2+3.4.4^2+3.4.5^2}\)

\(=\dfrac{1.2.\left(1+2^2+3^2+4^2+5^2\right)}{3.4.\left(1+2^2+3^2+4^2+5^2\right)}\)

\(=\dfrac{1.2}{3.4}=\dfrac{1}{6}\)

12:

\(S=\dfrac{1}{2}+\dfrac{2}{4}+...+\dfrac{10}{2^{10}}\)

\(=\dfrac{1}{2^1}+\dfrac{2}{2^2}+...+\dfrac{10}{2^{10}}\)

\(=\dfrac{1+1}{2^{1-1}}-\dfrac{1+2}{2^1}+\dfrac{2+1}{2^{2-1}}-\dfrac{2+2}{2^2}+...+\dfrac{10+1}{2^{10-1}}-\dfrac{10+2}{2^{10}}\)

\(=2-\dfrac{12}{2^{10}}=\dfrac{2^{11}-12}{2^{10}}\)

\(\dfrac{x-1}{2}=\dfrac{2-x}{3}\)

\(\Rightarrow3\cdot\left(x-1\right)=2\cdot\left(2-x\right)\)

\(\Rightarrow3x-3=4-2x\)

\(\Rightarrow3x+2x=4+3\)

\(\Rightarrow5x=7\)

\(\Rightarrow x=\dfrac{7}{5}\)

\(x-\dfrac{1}{2}=2-\dfrac{x}{3}\)

\(x+\dfrac{x}{3}=2+\dfrac{1}{2}\)

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

\(x\times\dfrac{4}{3}=\dfrac{5}{2}\)

\(x=\dfrac{5}{2}:\dfrac{4}{3}\)

\(x=\dfrac{15}{8}\)

Câu 6:

a: MB=2MC

=>\(\dfrac{CM}{CB}=\dfrac{1}{3};\dfrac{BM}{BC}=\dfrac{2}{3}\)

Ta có: BM=2/3BC

=>\(S_{MBE}=\dfrac{2}{3}\cdot S_{BEC}\)

Vì CM=1/3CB

nên \(S_{MCD}=\dfrac{1}{3}\cdot S_{BCD}\)

Xét ΔMAB và ΔMEC có

\(\widehat{MAB}=\widehat{MEC}\)(hai góc so le trong, AB//EC)

\(\widehat{AMB}=\widehat{EMC}\)(hai góc đối đỉnh)

Do đó: ΔMAB~ΔMEC

=>\(\dfrac{AB}{EC}=\dfrac{BM}{CM}=2\)

=>\(\dfrac{CD}{CE}=2\)

=>\(S_{BCD}=2\cdot S_{BCE}\)

=>\(\dfrac{1}{3}\cdot S_{BCD}=\dfrac{2}{3}\cdot S_{BCE}\)

=>\(S_{MBE}=S_{MCD}\)

b: \(MB=\dfrac{2}{3}BC\)

mà BC=AD

nên \(\dfrac{MB}{AD}=\dfrac{2}{3}\)

Xét ΔOBM và ΔODA có

\(\widehat{OBM}=\widehat{ODA}\)(BM//DA)

\(\widehat{BOM}=\widehat{DOA}\)(hai góc đối đỉnh)

Do đó: ΔOBM~ΔODA

=>\(\dfrac{OB}{OD}=\dfrac{MB}{DA}=\dfrac{2}{3}\)

Bài 43: Sửa đề: Trên tia Ox

a: Trên tia Ox, ta có: OA<OB

nên A nằm giữa O và B

=>OA+AB=OB

=>AB+4=7

=>AB=3(cm)

b: Vì BA và BD là hai tia đối nhau

nên B nằm giữa A và D

=>AD=AB+BD=3+1=4(cm)

c: Vì OA=AD(=4cm)

nên A là trung điểm của OD

Bài 44:

\(S=\dfrac{3^2}{1\cdot3}+\dfrac{3^2}{3\cdot5}+...+\dfrac{3^2}{2021\cdot2023}\)

\(=\dfrac{9}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{2021\cdot2023}\right)\)

\(=\dfrac{9}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2021}-\dfrac{1}{2023}\right)\)

\(=\dfrac{9}{2}\left(1-\dfrac{1}{2023}\right)=\dfrac{9}{2}\cdot\dfrac{2022}{2023}=\dfrac{9099}{2023}\)

\(\dfrac{11}{19}.\dfrac{12}{29}-\dfrac{11}{19}.\dfrac{2}{29}+\dfrac{11}{19}.\dfrac{19}{29}\)

\(=\dfrac{11}{19}.\left(\dfrac{12}{29}-\dfrac{2}{29}+\dfrac{19}{29}\right)\)

\(=\dfrac{11}{19}.1\)

\(=\dfrac{11}{19}\)

3/4 giá niêm yết là:

\(\dfrac{3}{4}\cdot300000=225000\left(đồng\right)\)

Giá vốn là \(225000\cdot\dfrac{100}{125}=180000\left(đồng\right)\)

Để lãi 40% so với giá vốn thì giá tiền cửa hàng cần bán là:

\(180000\left(1+40\%\right)=180000\cdot1,4=252000\left(đồng\right)\)

\(29\cdot5^{2024}-3\cdot25^x=14\cdot5^{2024}\)

=>\(3\cdot5^{2x}=29\cdot5^{2024}-14\cdot5^{2024}=15\cdot5^{2024}=3\cdot5^{2025}\)

=>2x=2025

=>\(x=\dfrac{2025}{2}\)