To solve the inequality [tex]\( -2(7y - 7) + y > 2y - (-5 + y) \)[/tex], let's work through the problem step-by-step.
1. Distribute the constants inside the parentheses:
[tex]\[
-2(7y - 7) + y > 2y - (-5 + y)
\][/tex]
Applying the distributive property:
[tex]\[
-2 \cdot 7y + (-2) \cdot (-7) + y > 2y - (-5) + (-1) \cdot y
\][/tex]
Simplify the terms:
[tex]\[
-14y + 14 + y > 2y + 5 - y
\][/tex]
2. Combine like terms on both sides of the inequality:
On the left side:
[tex]\[
-14y + y + 14 = -13y + 14
\][/tex]
On the right side:
[tex]\[
2y - y + 5 = y + 5
\][/tex]
So the inequality becomes:
[tex]\[
-13y + 14 > y + 5
\][/tex]
3. Isolate the variable terms on one side:
Subtract [tex]\( y \)[/tex] from both sides:
[tex]\[
-13y - y + 14 > 5
\][/tex]
Simplify:
[tex]\[
-14y + 14 > 5
\][/tex]
4. Isolate the constant term:
Subtract 14 from both sides:
[tex]\[
-14y > 5 - 14
\][/tex]
Simplify:
[tex]\[
-14y > -9
\][/tex]
5. Solve for the variable by dividing both sides by -14. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
y < \frac{-9}{-14}
\][/tex]
Simplify the fraction:
[tex]\[
y < \frac{9}{14}
\][/tex]
6. Write the solution set in interval notation:
[tex]\[
(-\infty, \frac{9}{14})
\][/tex]
Thus, the correct answer is:
c. [tex]\(\left(-\infty, \frac{9}{14}\right)\)[/tex]
