Solve for x:
(x + 1)/2 + (2 x - 3)/9 = 11
Put each term in (x + 1)/2 + (2 x - 3)/9 over the common denominator 18: (x + 1)/2 + (2 x - 3)/9 = (9 (x + 1))/18 + (2 (2 x - 3))/18:
(9 (x + 1))/18 + (2 (2 x - 3))/18 = 11
(9 (x + 1))/18 + (2 (2 x - 3))/18 = (9 (x + 1) + 2 (2 x - 3))/18:
(9 (x + 1) + 2 (2 x - 3))/18 = 11
9 (x + 1) = 9 x + 9:
(9 x + 9 + 2 (2 x - 3))/18 = 11
2 (2 x - 3) = 4 x - 6:
(4 x - 6 + 9 x + 9)/18 = 11
Grouping like terms, 9 x + 4 x - 6 + 9 = (9 x + 4 x) + (9 - 6):
((9 x + 4 x) + (9 - 6))/18 = 11
9 x + 4 x = 13 x:
(13 x + (9 - 6))/18 = 11
9 - 6 = 3:
(13 x + 3)/18 = 11
Multiply both sides of (13 x + 3)/18 = 11 by 18:
(18 (13 x + 3))/18 = 18×11
(18 (13 x + 3))/18 = 18/18×(13 x + 3) = 13 x + 3:
13 x + 3 = 18×11
18×11 = 198:
13 x + 3 = 198
Subtract 3 from both sides:
13 x + (3 - 3) = 198 - 3
3 - 3 = 0:
13 x = 198 - 3
198 - 3 = 195:
13 x = 195
Divide both sides of 13 x = 195 by 13:
(13 x)/13 = 195/13
13/13 = 1:
x = 195/13
The gcd of 195 and 13 is 13, so 195/13 = (13×15)/(13×1) = 13/13×15 = 15:
Answer: |x = 15
