To solve for [tex]\( x \)[/tex] in the equation
[tex]\[
\frac{7x + 14}{3} - \frac{17 - 3x}{5} = 6x - \frac{4x + 2}{3} - 5,
\][/tex]
let us go through the solution step-by-step:
### Step 1: Simplify Both Sides Separately
Rewrite the equation for clarity:
[tex]\[
\frac{7x + 14}{3} - \frac{17 - 3x}{5} = 6x - \frac{4x + 2}{3} - 5.
\][/tex]
First, we combine the fractions on both sides:
### Step 2: Combine Like Terms on Each Side
For the left side:
[tex]\[
\frac{7x + 14}{3} - \frac{17 - 3x}{5}
\][/tex]
Get a common denominator for the fractions:
[tex]\[
\frac{5(7x + 14) - 3(17 - 3x)}{15}
\][/tex]
[tex]\[
= \frac{35x + 70 - 51 + 9x}{15}
\][/tex]
Combine like terms:
[tex]\[
= \frac{44x + 19}{15}
\][/tex]
For the right side:
[tex]\[
6x - \frac{4x + 2}{3} - 5
\][/tex]
Get a common denominator for the expressions involving [tex]\( x \)[/tex]:
[tex]\[
= 6x - \frac{4x + 2}{3} - 5
\][/tex]
[tex]\[
= 6x - \frac{4x + 2}{3} - \frac{15}{3}
\][/tex]
Get a common denominator:
[tex]\[
= 6x - \frac{4x + 2 + 15}{3}
\][/tex]
Combine like terms:
[tex]\[
= 6x - \frac{4x + 17}{3}
\][/tex]
### Step 3: Set the Simplified Expressions Equal
Now equate the simplified expressions:
[tex]\[
\frac{44x + 19}{15} = 6x - \frac{4x + 17}{3}
\][/tex]
### Step 4: Eliminate Denominators
Multiply both sides by 15 to eliminate the fraction:
[tex]\[
44x + 19 = 15 \left(6x - \frac{4x + 17}{3}\right)
\][/tex]
Distribute 15 inside the parenthesis:
[tex]\[
= 15 \times 6x - 15 \times \frac{4x + 17}{3}
\][/tex]
Distribute within the fraction:
[tex]\[
= 90x - 5(4x + 17)
\][/tex]
Simplify within the fraction:
[tex]\[
= 90x - 20x - 85
\][/tex]
Combine like terms:
[tex]\[
= 70x - 85
\][/tex]
Therefore, the equation becomes:
[tex]\[
44x + 19 = 70x - 85
\][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], first move all [tex]\( x \)[/tex]-terms to one side and constants to the other:
[tex]\[
44x - 70x = -85 - 19
\][/tex]
Simplify:
[tex]\[
-26x = -104
\][/tex]
Divide both sides by -26:
[tex]\[
x = 4
\][/tex]
### Conclusion
The value of [tex]\( x \)[/tex] that satisfies the given equation is:
[tex]\[
\boxed{4}
\][/tex]
