sqrt(8-3x)+sqrt(6x-1)=7-(2*(sqrt(9x-24)))
are we solving
√(8-3x) + √(6x-1) = 7 - 2√(9x-24) ????
I tested it in Wolfram, and there is no "real" solution, so I won't even try.
http://www.wolframalpha.com/input/?i=%E2%88%9A(8-3x)+%2B+%E2%88%9A(6x-1)+%3D+7+-+2%E2%88%9A(9x-24)
notice that the left side is only defined for
8-3x ≥ 0 AND 6x-1 ≥0
-3x ≥ -8 AND 6x ≥ 1
x ≤ 8/3 AND x ≥ 1/6
1/6 ≤ x ≤ 8/3 ----> the blue part of the graph
on the right side we have:
9x-24 ≥ 0
9x ≥ 24
x ≥ 8/3 , the only common value of the domains
if x = 8/3
RS = 7 - 2√0 = 7
LS = √0 + √15 ≠ 7
NO REAL SOLUTION!
To solve the equation sqrt(8-3x) + sqrt(6x-1) = 7 - 2√(9x-24), we can follow these steps:
Step 1: Isolate one of the square roots on one side of the equation.
Subtract 7 from both sides of the equation:
sqrt(8-3x) + sqrt(6x-1) - 7 = -2√(9x-24)
Step 2: Square both sides of the equation to get rid of the remaining square root.
Squaring both sides will help eliminate the square roots:
(sqrt(8-3x) + sqrt(6x-1) - 7)^2 = (-2√(9x-24))^2
(8-3x) + 2(sqrt(8-3x) * sqrt(6x-1)) - 14(sqrt(8-3x)) - 14(sqrt(6x-1)) + (6x-1) = 4(9x-24)
Step 3: Simplify the equation.
Expand both sides and combine like terms:
8 - 3x + 2(sqrt(8-3x) * sqrt(6x-1)) - 14(sqrt(8-3x)) - 14(sqrt(6x-1)) + 6x - 1 = 36x - 96
Step 4: Isolate the remaining square root terms on one side of the equation.
Rearrange the terms to isolate the square root terms:
-3x + 6x + 8 - 1 - 36x + 14(sqrt(8-3x)) + 14(sqrt(6x-1)) - 2(sqrt(8-3x) * sqrt(6x-1)) = -96
Step 5: Combine the x terms and the constant terms.
Combine the x terms and the constant terms on both sides of the equation:
-33x + 7(sqrt(8-3x)) + 14(sqrt(6x-1)) - 2(sqrt(8-3x) * sqrt(6x-1)) = -96 - 7 + 1
Step 6: Simplify the equation further.
Simplify the right side of the equation:
-33x + 7(sqrt(8-3x)) + 14(sqrt(6x-1)) - 2(sqrt(8-3x) * sqrt(6x-1)) = -102
The equation is now simplified, but there is no clear way to solve it algebraically. You may need to use numerical methods or a calculator to approximate the solutions.