To solve the equation 9|3−8x|=2x+3, we need to consider two cases separately when the expression inside the absolute value is positive and when it is negative.
Case 1: When 3 - 8x is positive
In this case, the equation can be written as 9(3 - 8x) = 2x + 3.
Distribute 9: 27 - 72x = 2x + 3.
Combine like terms: -72x - 2x = 3 - 27.
Simplify: -74x = -24.
Divide by -74: x = (-24) / (-74).
Simplify: x = 12 / 37.
Case 2: When 3 - 8x is negative
In this case, we need to negate the expression inside the absolute value and solve for x.
The equation becomes 9(-(3 - 8x)) = 2x + 3.
Distribute the negative sign: 9(-3 + 8x) = 2x + 3.
Simplify: -27 + 72x = 2x + 3.
Combine like terms: 72x - 2x = 3 + 27.
Simplify: 70x = 30.
Divide by 70: x = 30 / 70.
Simplify: x = 3 / 7.
So the two solutions are x = 12/37 and x = 3/7.
To check for extraneous solutions, substitute these values back into the original equation:
For x = 12/37:
9|3 - 8(12/37)| = 2(12/37) + 3
9|(111 - 96)/37| = 24/37 + 3
9|15/37| = 24/37 + 3
9(15/37) = 24/37 + 3
135/37 = (24 + 111)/37
135/37 = 135/37
The left side is equal to the right side, so x = 12/37 is a valid solution.
For x = 3/7:
9|3 - 8(3/7)| = 2(3/7) + 3
9|(21 - 24)/7| = 6/7 + 3
9|-3/7| = 6/7 + 3
9(3/7) = 6/7 + 21/7
27/7 = 27/7
Again, the left side is equal to the right side, so x = 3/7 is also a valid solution.
Therefore, the solutions to the equation 9|3−8x|=2x+3 are x = 12/37 and x = 3/7.
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Solve the equation. Check for extraneous solutions.
9|3−8x|=2x+3
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