Answer: 11) x = 1 13) [tex]\bold{11)\quad x=1\qquad 13)\quad \dfrac{4}{9}\qquad 15)\quad \dfrac{7}{45}\qquad 17)\quad \dfrac{5}{12}}[/tex]
Step-by-step explanation:
11) Multiply everything by the LCD to clear the denominator, then distribute the negative sign.
[tex]\dfrac{2(2x+1)}{5}-\dfrac{x+2}{3}=\dfrac{1}{5}\\\\\\(15)\dfrac{2(2x+1)}{5}-(15)\dfrac{x+2}{3}=(15)\dfrac{1}{5}\\\\\\3\cdot2(2x+1)-5(x+2)=3(1)\\\\\\6(2x+1)-5(x+2)=3(1)\\\\12x + 6 - (5x + 10) = 3\\\\12x + 6 - 5x - 10 = 3\\\\7x-4=3\\\\7x=7\\\\\large\boxed{x=1}[/tex]
13) Since the repeating digit is in the tenths place, divide the repeating decimal by 9
[tex]0.\overline4\quad =\dfrac{4}{9}[/tex]
15) Since the repeating digit is in the hundredths place, divide the repeating decimal by 90 and divide the non-repeating decimal by 10
[tex]0.1\overline5\quad =\dfrac{1}{10}+\dfrac{5}{90}\\\\\\.\quad \quad =\bigg(\dfrac{9}{9}\bigg)\dfrac{1}{10}+\dfrac{5}{90}\\\\\\.\quad \quad =\dfrac{9}{90}+\dfrac{5}{90}\\\\\\.\quad \quad =\dfrac{14}{90}\qquad \text{simplify by dividing by 2}\\\\.\quad \quad =\large\boxed{\dfrac{7}{45}}[/tex]
17) Since the repeating digit is in the thousandths place, divide the repeating decimal by 900 and the non-repeating decimal by 100
[tex]0.41\overline6\quad =\dfrac{41}{100}+\dfrac{6}{900}\\\\\\.\quad \quad =\bigg(\dfrac{9}{9}\bigg)\dfrac{41}{100}+\dfrac{6}{900}\\\\\\.\quad \quad =\dfrac{369}{900}+\dfrac{6}{900}\\\\\\.\quad \quad =\dfrac{375}{900}\qquad \text{simplify by dividing by 75}\\\\\\.\quad \quad =\large\boxed{\dfrac{5}{12}}[/tex]
