To solve the equation, we'll start by removing the absolute value sign.
Case 1: 9 - 8x ≥ 0 (positive value inside the absolute value)
9 - 8x = 2x + 3
Let's solve for x:
9 - 8x = 2x + 3
-8x - 2x = 3 - 9
-10x = -6
x = (-6)/(-10)
x = 3/5
Case 2: 9 - 8x < 0 (negative value inside the absolute value)
-(9 - 8x) = 2x + 3
Let's solve for x:
-(9 - 8x) = 2x + 3
-9 + 8x = 2x + 3
8x - 2x = 3 + 9
6x = 12
x = 12/6
x = 2
Therefore, the solution to the equation is x = 3/5 and x = 2.
To check for extraneous solutions, we substitute both values back into the original equation:
For x = 3/5:
9|9 - 8(3/5)| = 2(3/5) + 3
9|9 - 24/5| = 6/5 + 3
9|45/5 - 24/5| = 6/5 + 3
9|21/5| = 21/5 + 15/5
9(21/5) = 36/5
189/5 = 36/5
The equation is false, so x = 3/5 is an extraneous solution.
For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 7
9(7) = 7
The equation is true, so x = 2 is a valid solution.
Therefore, the only solution to the equation is x = 2.
Include the 9 in front of the absolute value sign into this equation.
To include the 9 in front of the absolute value sign, we'll modify the equation as follows:
9|9 - 8x| = 2x + 3
Now, we'll continue solving the equation using the same steps as before.
Case 1: 9 - 8x ≥ 0 (positive value inside the absolute value)
9 - 8x = 2x + 3
Let's solve for x:
9 - 8x = 2x + 3
-8x - 2x = 3 - 9
-10x = -6
x = (-6)/(-10)
x = 3/5
Case 2: 9 - 8x < 0 (negative value inside the absolute value)
-(9 - 8x) = 2x + 3
Let's solve for x:
-(9 - 8x) = 2x + 3
-9 + 8x = 2x + 3
8x - 2x = 3 + 9
6x = 12
x = 12/6
x = 2
Now, let's check if the solutions are valid or extraneous:
For x = 3/5:
9|9 - 8(3/5)| = 2(3/5) + 3
9|9 - 24/5| = 6/5 + 3
9|-6/5| = 6/5 + 3
9(6/5) = 36/5
54/5 = 36/5
The equation is false, so x = 3/5 is an extraneous solution.
For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 7
9(7) = 7
The equation is true, so x = 2 is a valid solution.
Therefore, the only solution to the equation with the 9 included is x = 2.
To include the 9 in front of the absolute value sign, let's rewrite the equation in both cases:
Case 1: 9 - |9 - 8x| ≥ 0
9 - (9 - 8x) = 2x + 3
Simplifying:
9 - 9 + 8x = 2x + 3
8x = 2x + 3
6x = 3
x = 1/2
Case 2: 9 - |9 - 8x| < 0
-(9 - 8x) = 2x + 3
Simplifying:
-9 + 8x = 2x + 3
6x = 12
x = 2
Therefore, the solution to the equation 9 - |9 - 8x| ≥ 0 is x = 1/2, and the solution to the equation 9 - |9 - 8x| < 0 is x = 2.
To include the 9 in front of the absolute value sign, we will modify the equation as follows:
Case 1: 9 - |9 - 8x| ≥ 0 (positive value inside the absolute value)
9 - (9 - 8x) = 2x + 3
Let's solve for x:
9 - (9 - 8x) = 2x + 3
9 - 9 + 8x = 2x + 3
8x = 2x + 3
8x - 2x = 3
6x = 3
x = 3/6
x = 1/2
Case 2: 9 - |9 - 8x| < 0 (negative value inside the absolute value)
-(9 - 8x) = 2x + 3
Let's solve for x:
-(9 - 8x) = 2x + 3
-9 + 8x = 2x + 3
8x - 2x = 3 + 9
6x = 12
x = 12/6
x = 2
Therefore, the solution to the equation when including the 9 in front of the absolute value sign is x = 1/2 and x = 2.
To check for extraneous solutions, we substitute both values back into the original equation:
For x = 1/2:
9 - |9 - 8(1/2)| = 2(1/2) + 3
9 - |9 - 4| = 1 + 3
9 - |5| = 4
9 - 5 = 4
The equation is true, so x = 1/2 is a valid solution.
For x = 2:
9 - |9 - 8(2)| = 2(2) + 3
9 - |9 - 16| = 4 + 3
9 - |-7| = 7
9 - 7 = 7
The equation is true, so x = 2 is a valid solution.
Therefore, the solutions to the equation when including the 9 in front of the absolute value sign are x = 1/2 and x = 2.