Practice Problems

1. \(-\dfrac{9}{8}\;+\;{\text{g}}=-\dfrac{3}{8}\) (
   Solution

   x

   Solution: \(\dfrac{3}{4}\)
   Details:
   In this example, we are solving for the variable g. The only operation involved between g and other numbers is addition. We solve this the same way we solve equations with integers. We unravel the equation using the order of operations backward.
   
   Step 1: Do the inverse of any addition or subtraction.
   
   On the left-hand side of the equal sign \(-\dfrac{9}{8}\) is being added to g. We can remove this by adding the inverse to both sides of the equation.
   
   Add \(+\dfrac{9}{8}\) to both sides.
   
   
   The left side of the equation:
   \(-\dfrac{9}{8}+\dfrac{9}{8}=0\) leaving just the variable g.
   
   The right side of the equation:
   
   Since both of these fractions have the same denominator we can add the numerators.
   
   \(-\dfrac{3}{8}+\dfrac{9}{8}=\dfrac{6}{8}\)
   
   
   
   Normally this would be the end of our work since we isolated the solution for variable g, but in this case, we can still simplify our fraction.
   
   6 and 8 both have 2 as a common factor.
   
   \(6=(2)( 3)\)
   
   \(8=(2)(4)\)
   
   We can divide out the 2’s leaving just \(\dfrac{3}{4}\).
   
   \({\text{g}}=\dfrac{6}{8}=\dfrac{\left ( 2 \right )\left ( 3 \right )}{\left ( 2 \right )\left ( 4 \right )}=\dfrac{{\color{Red} \cancel{\left ( 2 \right )} }\left ( 3 \right )}{{\color{Red} \cancel{\left ( 2 \right )}}\left ( 4 \right )}=\dfrac{3}{4}\)
   
   Our final solution: \({\text{g}} = \dfrac{3}{4}\)
   )
2. \({\text{r}}+\dfrac{1}{4}=-1\) (
   Solution

   x

   Solution:
   \(-\dfrac{5}{4}\)
   )
3. \({\text{x}}+\dfrac{5}{9}=\dfrac{1}{3}\) (
   Solution

   x

   Solution: \(-\dfrac{2}{9}\)
   Details:
   In this example, we need to get the variable x all alone. To do this we add the inverse of \(\dfrac{5}{9}\) to both sides of the equation.
   
   We add \(-\dfrac{5}{9}\) to both sides.
   
   
   
   The left side of the equation:
   
   Positive \(\dfrac{5}{9}\) plus negative \(\dfrac{5}{9}\) equals 0. This leaves just the variable x by itself.
   
   
   
   The right side of the equation:
   
   In order to add \(\dfrac{1}{3}+-\dfrac{5}{9}\) we need to first get common denominators.
   
   9 is a multiple of 3, so our least common multiple is 9.
   
   Multiply \(\dfrac{1}{3}\) by 1 in the form of \(\dfrac{3}{3}\) to change its denominator to 9.
   
   \({\text{X}}=\left (\dfrac{1}{3} \right ){\color{Cyan} \left (\dfrac{3}{3} \right )}{\color{DarkGreen} + {\color{Red} -}\dfrac{5}{9}}\)
   
   \({\text{X}}=\dfrac{1 \cdot {\color{Cyan} 3}}{3 \cdot {\color{Cyan} 3}}{\color{DarkGreen} + {\color{Red} -}\dfrac{5}{9}}\)
   
   \({\text{X}}={\color{Cyan} \dfrac{3}{9}} {\color{DarkGreen} + {\color{Red} -}\dfrac{5}{9}}\)
   
   Now that our fractions have the same denominator, we can add the numerators.
   
   \(3+-5=-2\)
   
   \({\text{X}}=\dfrac{{\color{Cyan} 3}}{9} + {\color{Red} -}\dfrac{{\color{DarkGreen} 5}}{9}\)
   
   \({\text{X}}=\dfrac{{\color{Cyan} 3+{\color{Red} -}{\color{DarkGreen} 5}}}{9}\)
   
   \({\text{X}}=\dfrac{{\color{Red} -}{\color{Orange} 2}}{9}\)
   
   Our final solution: \({\text{X}} = -\dfrac{2}{9}\)
   )
4. \(-3+{\text{g}}=\dfrac{5}{2}\) (
   Solution

   x

   Solution:
   \(\dfrac{11}{2}\)
   )
5. \(-4={\text{j}}+\dfrac{4}{3}\) (
   Video Solution

   x

   Solution: \(\dfrac{-16}{3}\)
   Details:
   

   
   (Video Source | Transcript)
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6. \(\dfrac{7}{6}={\text{A}}-\dfrac{1}{2}\) (
   Solution

   x

   Solution:
   \(\dfrac{5}{3}\)
   )
7. \(\dfrac{-9}{8}=\dfrac{-3}{4}+{\text{a}}\) (
   Video Solution

   x

   Solution: \(\dfrac{-3}{8}\)
   Details:
   

   
   (Video Source | Transcript)
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