Final answer:
To solve the equation 4|x-8|-8 = -5|x-8|+19, isolate the absolute value terms on both sides, combine like terms, distribute the coefficients, and solve for x by dividing both sides by -9. The solutions are x = 11 and x = 5.
Step-by-step explanation:
To solve the equation 4|x-8|-8 = -5|x-8|+19, we can start by isolating the absolute value terms on each side of the equation. To do this, we can divide the entire equation by -1 to change the signs, resulting in -4|x-8|+8 = 5|x-8|-19.
Next, we can combine like terms on both sides: -4|x-8|+8 = 5|x-8|-19 can be simplified to -4|x-8| - 5|x-8| = -27.
Then, we can distribute the coefficients: -9|x-8| = -27.
Finally, we can solve for x by dividing both sides of the equation by -9, resulting in |x-8| = 3. This equation has two solutions: x-8 = 3 and x-8 = -3. Solving for x in each case gives x = 11 and x = 5.