square root>(3x + 1)− square root>(7x + 8)= 1

√(3x+1) - √(7x+8) = 1 , clearly x≥-1/3

√(3x+1) = 1 + √(7x+8)
square both sides

3x+1 = 1 + 2√(7x+8) + 7x+8
-4x -8 = 2√(7x+8)
-2x-4 = √7x+8)
square both sides again
4x^2 + 16x + 16 = 7x+8
4x^2 + 9x + 8 = 0
x = (-9 ± √-47)/8

no real solution.

I'm sorry I wrote the equation wrong 3�ã(x+1)-�ã(7x+8)=1

square roots to replace the a's

so

3√(x+1) - √(7x+8) = 1 , clearly x≥-1
3√(x+1) = 1 + √(7x+8)
square both sides

9(x+1) = 1 + 2√(7x+8) + 7x+8
9x+9 = 1 + 2√(7x+8) + 7x+8
2x = 2√(7x+8)
square again
4x^2= 4(7x+8)
4x^2 - 28x - 32 = 0
x^2 - 7x - 8 = 0
(x-8)(x+1) = 0
x=8 or x=-1

since we squared, all answers must be checked

if x = 8
LS = 3√9 - √(7(8) + 8)
= 9 - 8
= RS

if x = -1
LS = 3√0 - √(-7)+8))
= 0-1
≠ RS

So x = 8

To solve the equation square root>(3x + 1) − square root>(7x + 8) = 1, we need to isolate the variable x.

Step 1: Begin by isolating one of the square roots.
square root>(3x + 1) = 1 + square root>(7x + 8)

Step 2: Next, square both sides of the equation to eliminate the square root on the left side.
(square root>(3x + 1))^2 = (1 + square root>(7x + 8))^2

Step 3: Simplify both sides of the equation.
3x + 1 = 1 + 2 * square root>(7x + 8) + (7x + 8)

Step 4: Combine like terms.
3x + 1 = 2 * square root>(7x + 8) + 7x + 9

Step 5: Subtract 1 from both sides of the equation.
3x = 2 * square root>(7x + 8) + 7x + 8

Step 6: Move all the terms containing x to one side and all the constant terms to the other side.
3x - (7x + 8) = 2 * square root>(7x + 8)

Step 7: Simplify the expression on the left side.
3x - 7x - 8 = 2 * square root>(7x + 8)

Step 8: Combine like terms.
-4x - 8 = 2 * square root>(7x + 8)

Step 9: Divide both sides of the equation by 2.
(-4x - 8) / 2 = square root>(7x + 8)

Step 10: Simplify the expression on the right side.
-2x - 4 = square root>(7x + 8)

Step 11: Square both sides of the equation to eliminate the square root.
(-2x - 4)^2 = (square root>(7x + 8))^2

Step 12: Simplify both sides of the equation.
4x^2 + 16x + 16 = 7x + 8

Step 13: Move all terms to one side of the equation to obtain a quadratic equation.
4x^2 + 16x + 16 - 7x - 8 = 0

Step 14: Simplify the equation.
4x^2 + 9x + 8 = 0

Now, you can use various methods to solve this quadratic equation, such as factoring, completing the square, or using the quadratic formula.