Solve. Check for extraneous solutions.

1) (/3x+7) = x-1

2) (3/2x)-3=9

Note: (/3x+7) is the square root of 3x plus 7 and (3/2x)-3=9 is 3 times the square root of 2x - 3 equals 9.

1. sqrt(3x+7) = x-1.

Square both sides:
3x + 7 = x^2 - 2x + 1.
-x^2 + 2x +3x = 1 - 7 = - 6.
Multiply both sides by -1:
x^2 -2x -3x = 6.
x^2 - 5x - 6 = 0.
(x+1)(x-6) = 0.

x+1 = 0.
X = -1. Does not satisfy original Eq.

x-6 = 0.
X = 6.

2. 3*sqrt(2x) - 3 = 9.
3*sqrt(2x) = 9+3 = 12.
Divide both sides by 3:
sqrt(2x) = 4.
Square both sides:
2x = 16.
X = 8.

To solve these equations and check for extraneous solutions, follow these steps:

1) (/3x+7) = x-1

Step 1: Square both sides of the equation to remove the square root.

(√(3x + 7))^2 = (x - 1)^2

Step 2: Simplify both sides of the equation.

3x + 7 = x^2 - 2x + 1

Step 3: Rearrange the equation to form a quadratic equation.

x^2 - 5x - 6 = 0

Step 4: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.

(x - 6)(x + 1) = 0
x = 6 or x = -1

Step 5: Check for extraneous solutions by plugging both solutions back into the original equation.

Checking x = 6:
(√(3(6) + 7)) = 6 - 1
√(25) = 5
5 = 5 (True)

Checking x = -1:
(√(3(-1) + 7)) = -1 - 1
√(4) = -2
2 = -2 (False)

Extraneous Solution: x = -1

2) (3/2x)-3=9

Step 1: Add 3 to both sides of the equation to isolate the radical term.

(3/2x) = 12

Step 2: Multiply both sides of the equation by 2x to get rid of the fraction.

3 = 24x

Step 3: Divide both sides of the equation by 24 to solve for x.

x = 3/8

Step 4: Check for extraneous solutions by plugging the value of x back into the original equation.

Checking x = 3/8:
(3/2(3/8)) - 3 = 9
(3/16) - 3 = 9
3 - 48 = 144 (False)

Extraneous Solution: x = 3/8

Therefore, the solution for the first equation is x = 6 (no extraneous solution), and the solution for the second equation does not exist (extraneous solution).