To solve the absolute value equation [tex]\( |14x + 3| + 5 = 18 \)[/tex], we follow these steps:
1. Isolate the absolute value expression:
[tex]\[
|14x + 3| + 5 = 18
\][/tex]
Subtract 5 from both sides:
[tex]\[
|14x + 3| = 13
\][/tex]
2. Set up the two possible equations for the absolute value:
The absolute value equation [tex]\( |A| = B \)[/tex] means [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Here, [tex]\( A = 14x + 3 \)[/tex] and [tex]\( B = 13 \)[/tex]. Thus, we write two equations:
[tex]\[
14x + 3 = 13
\][/tex]
and
[tex]\[
14x + 3 = -13
\][/tex]
3. Solve for [tex]\( x \)[/tex] in each case:
- For [tex]\( 14x + 3 = 13 \)[/tex]:
[tex]\[
14x + 3 = 13
\][/tex]
Subtract 3 from both sides:
[tex]\[
14x = 10
\][/tex]
Divide by 14:
[tex]\[
x = \frac{10}{14} = \frac{5}{7} \approx 0.7142857142857143
\][/tex]
- For [tex]\( 14x + 3 = -13 \)[/tex]:
[tex]\[
14x + 3 = -13
\][/tex]
Subtract 3 from both sides:
[tex]\[
14x = -16
\][/tex]
Divide by 14:
[tex]\[
x = \frac{-16}{14} = -\frac{8}{7} \approx -1.1428571428571428
\][/tex]
4. Sum the solutions:
We have [tex]\( x_1 = \frac{5}{7} \)[/tex] and [tex]\( x_2 = -\frac{8}{7} \)[/tex]. To find [tex]\( S \)[/tex], the sum of the solutions:
[tex]\[
S = \frac{5}{7} + (-\frac{8}{7}) = \frac{5}{7} - \frac{8}{7} = \frac{5 - 8}{7} = -\frac{3}{7} \approx -0.4285714285714285
\][/tex]
Therefore, the sum of the solutions [tex]\( S \)[/tex] is:
[tex]\[
S = -0.4285714285714285
\][/tex]
