To solve the equation
[tex]\[ \frac{1+2x}{2} + \frac{2-x}{3} = \frac{19}{6}, \][/tex]
we will go through it step-by-step to simplify and solve for [tex]\( x \)[/tex].
1. Identify and combine common denominators: We'll first combine the fractions on the left side of the equation over a common denominator.
[tex]\[
\frac{1+2x}{2} + \frac{2-x}{3}.
\][/tex]
The common denominator of 2 and 3 is 6. So, we'll rewrite each fraction with the common denominator of 6:
[tex]\[
\frac{(1+2x) \cdot 3}{6} + \frac{(2-x) \cdot 2}{6}.
\][/tex]
2. Simplify the fractions: Multiply out the numerators:
[tex]\[
\frac{3(1+2x)}{6} + \frac{2(2-x)}{6} = \frac{3 + 6x}{6} + \frac{4 - 2x}{6}.
\][/tex]
3. Combine the fractions: Combine the fractions into a single fraction:
[tex]\[
\frac{3 + 6x + 4 - 2x}{6} = \frac{7 + 4x}{6}.
\][/tex]
So the original equation now looks like:
[tex]\[
\frac{7 + 4x}{6} = \frac{19}{6}.
\][/tex]
4. Eliminate the denominators: Since both sides of the equation have the same denominator, we can multiply through by 6 to get rid of the denominators:
[tex]\[
7 + 4x = 19.
\][/tex]
5. Solve for [tex]\( x \)[/tex]: Isolate the variable [tex]\( x \)[/tex] by subtracting 7 from both sides:
[tex]\[
4x = 12.
\][/tex]
Then, divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3.
\][/tex]
Therefore, the solution to the equation is
[tex]\[ x = 3. \][/tex]
