To solve the equation given by the problem, we need to translate it into algebraic form and solve for the variable [tex]\(d\)[/tex].
1. Translate the problem into an equation:
The difference of [tex]\(-2d\)[/tex] and [tex]\(-3d\)[/tex] can be written as:
[tex]\[
\text{Difference of } -2d \text{ and } -3d = -2d - (-3d)
\][/tex]
The difference of [tex]\(\frac{1}{6}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex] can be written as:
[tex]\[
\text{Difference of } \frac{1}{6} \text{ and } \frac{1}{3} = \frac{1}{6} - \frac{1}{3}
\][/tex]
Now we set these two expressions equal to each other:
[tex]\[
-2d - (-3d) = \frac{1}{6} - \frac{1}{3}
\][/tex]
2. Simplify the left-hand side:
Simplifying the left-hand side:
[tex]\[
-2d - (-3d) = -2d + 3d = d
\][/tex]
3. Simplify the right-hand side:
Simplifying the right-hand side:
[tex]\[
\frac{1}{6} - \frac{1}{3}
\][/tex]
To subtract these fractions, we need a common denominator. The least common denominator for 6 and 3 is 6.
[tex]\[
\frac{1}{3} = \frac{2}{6}
\][/tex]
So,
[tex]\[
\frac{1}{6} - \frac{2}{6} = \frac{1 - 2}{6} = \frac{-1}{6}
\][/tex]
4. Equate both sides and solve for [tex]\(d\)[/tex]:
Now, we have:
[tex]\[
d = \frac{-1}{6}
\][/tex]
5. Check the solution:
Substitute [tex]\(d = \frac{-1}{6}\)[/tex] back into the equation to verify:
[tex]\[
-2 \left( \frac{-1}{6} \right) - (-3 \left( \frac{-1}{6} \right)) = \frac{1}{6} - \frac{1}{3}
\][/tex]
Simplifying the left-hand side:
[tex]\[
-2 \left( \frac{-1}{6} \right) + 3 \left( \frac{-1}{6} \right) = \frac{2}{6} - \frac{3}{6} = \frac{-1}{6}
\][/tex]
Both sides of the equation are equal, confirming that our solution is correct.
Thus, the solution is:
[tex]\[
d = -\frac{1}{6}
\][/tex]
