For which value of b would make the equation 3|x-2|=bx-6 have infinitely many solutions?
How would I find the values?
The choices are -6,-3,3,6.
To determine the value of b that would make the equation 3|x-2| = bx-6 have infinitely many solutions, we need to analyze the absolute value equation.
Step 1: Isolate the absolute value expression on one side of the equation:
3|x-2| = bx - 6
Step 2: Apply the definition of absolute value:
x - 2 = (bx - 6) / 3 OR x - 2 = -(bx - 6) / 3
Step 3: Simplify the equation in both cases:
x = (bx - 6)/3 + 2 OR x = -(bx - 6)/3 + 2
Step 4: Consider the first case:
x = (bx - 6)/3 + 2
Step 5: Multiply both sides of the equation by 3:
3x = bx - 6 + 6
Step 6: Simplify:
3x = bx
Step 7: Rearrange the equation:
bx - 3x = 0
Step 8: Factor out x:
x(b - 3) = 0
Step 9: Set each factor equal to zero to find possible values of x:
x = 0 or b - 3 = 0
Step 10: In the second case, consider:
x = -(bx - 6)/3 + 2
Step 11: Multiply both sides of the equation by 3:
3x = -(bx - 6) + 6
Step 12: Simplify:
3x = -bx + 6 + 6
Step 13: Rearrange the equation:
bx + 3x = 12
Step 14: Factor out x:
x(b + 3) = 12
Step 15: Set each factor equal to zero to find possible values of x:
x = 0 or b + 3 = 0
Step 16: Analyze the results:
From Step 9, we see that x can either be 0 or b - 3 = 0, which means b = 3.
From Step 15, we see that x can either be 0 or b + 3 = 0, which means b = -3.
So, the two values of b that would make the equation have infinitely many solutions are b = 3 and b = -3.
In this case, the answer would be the choice "b = 3 or b = -3".