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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'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.
To solve the equation 9|3−8x|=2x+3, we need to isolate the absolute value term and solve for x.
Step 1: Distribute the absolute value notation to the expression inside it:
9|3−8x| = 2x + 3
Step 2: Apply the definition of absolute value.
Since we have "|3-8x|" on the left side, we replace it with two separate equations:
1) 3 - 8x = 2x + 3 (when 3-8x > 0)
2) 3 - 8x = - (2x + 3) (when 3-8x < 0)
Step 3: Simplify both of the equations:
1) 3 - 8x = 2x + 3
Combine like terms:
-8x - 2x = 3 - 3
-10x = 0
Divide both sides by -10:
x = 0
2) 3 - 8x = -(2x + 3)
Distribute the negative sign:
3 - 8x = -2x - 3
Combine like terms:
-8x + 2x = -3 - 3
-6x = -6
Divide both sides by -6:
x = 1
Therefore, we have found two potential solutions: x = 0 and x = 1.
Step 4: Check for extraneous solutions.
To check if any of the solutions are extraneous, plug them back into the original equation and see if both sides are equal.
For x = 0:
9|3−8(0)| = 2(0) + 3
9|3| = 3
9(3) = 3
27 = 3
This is not true, so x = 0 is an extraneous solution.
For x = 1:
9|3−8(1)| = 2(1) + 3
9|3-8| = 2 + 3
9|-5| = 5
9(5) = 5
45 = 5
This is false, so x = 1 is also an extraneous solution.
Therefore, the equation 9|3−8x|=2x+3 has no valid solutions.