To solve the equation 9|3−8x|=2x+3, we need to remove the absolute value by separating it into two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: 3 - 8x > 0
Solving this inequality, we have:
3 > 8x
x < 3/8
Now, we substitute this solution into the original equation to check for extraneous solutions:
9|3−8x| = 2x + 3
9|(3−8(3/8))| = 2(3/8) + 3
9|3-3| = 6/8 + 3
9|0| = 6/8 + 3
0 = 6/8 + 3
0 = 27/4
Since the equation is not true, the solution x < 3/8 is extraneous.
Case 2: 3 - 8x < 0
Solving this inequality, we have:
3 < 8x
x > 3/8
Now, we substitute this solution into the original equation to check for extraneous solutions:
9|3−8x| = 2x + 3
9|(3−8(3/8))| = 2(3/8) + 3
9|3-3| = 6/8 + 3
9|0| = 6/8 + 3
0 = 6/8 + 3
0 = 27/4
Since the equation is not true, the solution x > 3/8 is extraneous.
Therefore, there are no valid solutions to the equation 9|3−8x|=2x+3.
9|3−8x|=2x+3 and check for extraneous solutions
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