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Solve the equation. Check for extraneous solutions.
9|3−8x|=2x+3
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.
To solve the equation, we need to remove the absolute value bars first. We have:
9|3 - 8x| = 2x + 3
We can start by dividing both sides of the equation by 9:
|3 - 8x| = (2x + 3)/9
Now we need to consider two cases:
Case 1: 3 - 8x ≥ 0 (No absolute value)
In this case, we can simply remove the absolute value bars without changing the equation:
3 - 8x = (2x + 3)/9
Next, we can multiply both sides of the equation by 9 to eliminate the fraction:
9(3 - 8x) = 2x + 3
27 - 72x = 2x + 3
Combine like terms:
-72x - 2x = 3 - 27
-74x = -24
Divide both sides of the equation by -74:
x = -24 / -74
Simplify the fraction:
x = 12 / 37
Case 2: 3 - 8x < 0 (Absolute value with negative sign)
In this case, we need to flip the inequality sign and change the absolute value expression to its negation:
-(3 - 8x) = (2x + 3)/9
Simplify the negation:
-3 + 8x = (2x + 3)/9
Next, we can multiply both sides of the equation by 9 to eliminate the fraction:
9(-3 + 8x) = 2x + 3
-27 + 72x = 2x + 3
Combine like terms:
72x - 2x = 3 + 27
70x = 30
Divide both sides of the equation by 70:
x = 30 / 70
Simplify the fraction:
x = 3 / 7
Now we need to check for extraneous solutions by substituting both values of x back into the original equation:
For x = 12 / 37:
9|3 - 8(12 / 37)| = 2(12 / 37) + 3
Simplifying:
9|3 - 96 / 37| = 24 / 37 + 111 / 37
Now evaluate the absolute value expression:
9|111 / 37 - 96 / 37| = 24 / 37 + 111 / 37
Simplify the fractions:
9|(111 - 96) / 37| = (24 + 111) / 37
Calculate the numerator inside the absolute value:
9|15 / 37| = 135 / 37
Since the absolute value of 15/37 is positive, we can remove the absolute value bars:
9 * 15 / 37 = 135 / 37
Simplify:
135 / 37 = 135 / 37
The left side equals the right side, so x = 12/37 is a valid solution.
For x = 3/7:
9|3 - 8(3 / 7)| = 2(3 / 7) + 3
Simplifying:
9|3 - 24 / 7| = 6 / 7 + 21 / 7
Now evaluate the absolute value expression:
9|21 / 7 - 24 / 7| = 6 / 7 + 21 / 7
Simplify the fractions:
9|(21 - 24) / 7| = (6 + 21) / 7
Calculate the numerator inside the absolute value:
9|-3 / 7| = 27 / 7
Since the absolute value of -3/7 is positive, we can remove the absolute value bars:
9 * 3 / 7 = 27 / 7
Simplify:
27 / 7 = 27 / 7
The left side equals the right side, so x = 3/7 is also a valid solution.
Therefore, the solutions to the equation are x = 12/37 and x = 3/7, and both solutions are valid (there are no extraneous solutions).