Letf (x) = x + |2x - 5|. Find all solutions to the equationf (x) = 10. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = (b) Letg (x) = 4x - 6 + |x + 3|. Find all values of a which satisfy the equationg (a) = 3a + 9. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) a =

Part 1 : Given equation,

f (x) = x + |2x - 5|,

We have,

f (x) = 10,

x + |2x - 5| = 10

|2x - 5| = 10 - x

2x - 5 = ± (10-x)

Case 1 : 2x - 5 = 10 - x

2x = 10 - x + 5

2x + x = 15

3x = 15

⇒ x = 5

Case 2 : 2x - 5 = - 10 + x

2x = - 10 + x + 5

2x - x = - 5

⇒ x = - 5

Thus, the value of x is 5 or - 5

Part 2 : Given equation,

g (x) = 4x - 6 + |x + 3|,

Put x = a,

g (a) = 4a - 6 + |a + 3|

We have, g (a) = 3a + 9

4a - 6 + |a+3|=3a + 9

4a + |a + 3 | = 3a + 9 + 6

|a + 3| = 3a - 4a + 15

|a + 3| = - a + 15

⇒ a + 3 = ± (-a + 15)

Case 1 : a + 3 = - a + 15

a = - a + 15 - 3

a + a = 12

2a = 12

⇒ a = 6

Case 2 : a + 3 = a - 15

a = a - 15 - 3

a - a = - 18

0 = - 18 (False)

Thus, the value of a is 6.