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Solving for a Variable on One Side Using Addition and Subtraction with Fractions
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When solving for variables with fractions instead of whole numbers, we still use the same process: adding the additive inverse.

The following video will show what the additive inverse of a fraction is and review some addition and subtraction of fractions:

Solving for a Variable on One Side Part 4-Addition and subtraction with fractions

As a rule, we can remember additive inverses as having the opposite sign (positive or negative) as our number. This is true with fractions as well.

Also remember, in order to add or subtract fractions, they must have a common denominator.

### Practice Problems

Solve for the variable:
1. $$-\dfrac{9}{8}\;+\;{\text{g}}=-\dfrac{3}{8}$$ (
Solution
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
Solution:
$$-\dfrac{5}{4}$$
)
3. $${\text{x}}+\dfrac{5}{9}=\dfrac{1}{3}$$ (
Solution
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
Solution:
$$\dfrac{11}{2}$$
)
5. $$-4={\text{j}}+\dfrac{4}{3}$$ (
Video Solution
Solution: $$\dfrac{-16}{3}$$
Details:

(Video Source | Transcript)
)
6. $$\dfrac{7}{6}={\text{A}}-\dfrac{1}{2}$$ (
Solution
Solution:
$$\dfrac{5}{3}$$
)
7. $$\dfrac{-9}{8}=\dfrac{-3}{4}+{\text{a}}$$ (
Video Solution
Solution: $$\dfrac{-3}{8}$$
Details:

(Video Source | Transcript)
)

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