9|3−8x|=2x+3
To solve this equation, we can start by isolating the absolute value on one side of the equation.
Let's begin by subtracting 3 from both sides:
9|3 - 8x| = 2x + 3 - 3
This simplifies to:
9|3 - 8x| = 2x
Next, we can divide both sides of the equation by 9:
|3 - 8x| = (2x) / 9
Now, we have two cases to consider:
Case 1: (3 - 8x) > 0
In this case, the absolute value is positive, so the equation becomes:
3 - 8x = (2x) / 9
To solve for x, let's start by multiplying both sides by 9:
27 - 72x = 2x
Next, let's add 72x to both sides:
27 = 74x
Finally, divide both sides by 74:
x = 27 / 74
Case 2: (3 - 8x) < 0
In this case, the absolute value is negative, so we need to flip the sign of the equation:
- (3 - 8x) = (2x) / 9
Let's distribute the negative sign:
-3 + 8x = (2x) / 9
Let's multiply both sides by 9 to eliminate the fraction:
-27 + 72x = 2x
Now, subtract 2x from both sides:
-27 + 70x = 0
Add 27 to both sides:
70x = 27
Finally, divide both sides by 70:
x = 27 / 70
So the solutions to the equation are:
x = 27 / 74 or x = 27 / 70
To solve the equation 9|3−8x|=2x+3, we will consider two cases depending on the value inside the absolute value bars.
Case 1: (3 - 8x) ≥ 0
In this case, we can remove the absolute value bars without changing the equation. Thus, we have:
9(3 - 8x) = 2x + 3
Simplifying the equation:
27 - 72x = 2x + 3
Bringing the x-terms to one side and the constant terms to the other side:
27 - 3 = 2x + 72x
24 = 74x
Dividing both sides by 74:
x = 24/74
This can be simplified to:
x = 12/37
So, one solution to the equation is x = 12/37.
Case 2: (3 - 8x) < 0
In this case, we need to flip the inequality when removing the absolute value bars. Thus, we have:
-9(3 - 8x) = 2x + 3
Simplifying the equation:
-27 + 72x = 2x + 3
Bringing the x-terms to one side and the constant terms to the other side:
72x - 2x = 3 + 27
70x = 30
Dividing both sides by 70:
x = 30/70
This can be simplified to:
x = 3/7
So, another solution to the equation is x = 3/7.
Therefore, the equation 9|3−8x|=2x+3 has two solutions: x = 12/37 and x = 3/7.