To solve the equation 9|3-8x| = 2x + 3, we'll need to isolate the absolute value expression and then solve for x.
First, we'll remove the absolute value by splitting the equation into two cases: one where the expression inside the absolute value is positive, and one where it's negative.
Case 1: 3 - 8x is positive.
9(3 - 8x) = 2x + 3
27 - 72x = 2x + 3
Combine like terms:
-72x - 2x = 3 - 27
-74x = -24
Divide both sides by -74 to solve for x:
x = (-24) / (-74)
Simplify the fraction (if possible):
x = 12 / 37
Case 2: 3 - 8x is negative.
We'll need to negate the expression inside the absolute value, and keep the equation inside the absolute value equal to the negative of the other side.
9(-3 + 8x) = 2x + 3
-27 + 72x = 2x + 3
Combine like terms:
70x = 3 + 27
70x = 30
Divide both sides by 70 to solve for x:
x = 30 / 70
Simplify the fraction (if possible):
x = 3 / 7
So, we have two potential solutions: x = 12/37 and x = 3/7.
To check for extraneous solutions, we substitute each solution back into the original equation and see if it holds true.
Checking x = 12/37:
9|3 - 8(12/37)| = 2(12/37) + 3
Simplify:
9|3 - 96/37| = 24/37 + 111/37
Combine like terms:
9|111/37 - 96/37| = 135/37
Simplify absolute value:
9|15/37| = 135/37
Both sides are positive, so the absolute value can be removed:
(9)(15/37) = 135/37
Simplify:
135/37 = 135/37
The equation holds true for x = 12/37.
Checking x = 3/7:
9|3 - 8(3/7)| = 2(3/7) + 3
Simplify:
9|3 - 24/7| = 6/7 + 21/7
Combine like terms:
9|21/7 - 24/7| = 27/7
Simplify absolute value:
9|-3/7| = 27/7
Both sides are positive, so the absolute value can be removed:
(9)(3/7) = 27/7
Simplify:
27/7 = 27/7
The equation also holds true for x = 3/7.
Therefore, the solutions x = 12/37 and x = 3/7 do not result in any extraneous solutions.
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Solve the equation. Check for extraneous solutions.
9|3−8x|=2x+3
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