when you are solving an equation involving absolute values, do you have to have both negative and positve solutions??

not necessarily, it depends on the equation.

e.g.#1
│2x-1│ = 5

then (2x-)=5 or -(2x-1)=5
x=3 or x=-2

here you have a positive as well as a negative solution.

e.g.#2

│11-2x│ = 5

(11-2x) = 5 or -(11-2x) = 5
x=3 or x=8

both solutions are positive.

When solving an equation involving absolute values, you may or may not have both negative and positive solutions. It depends on the specific equation you are working with.

To solve an equation involving absolute values, you need to consider two cases: one with the expression inside the absolute value being positive, and one with the expression being negative. This allows you to remove the absolute value and split the equation into two separate equations.

For example, let's consider the equation |2x-1| = 5.

In the first case, we assume 2x-1 is positive:
2x-1 = 5
Solving for x, we get x = 3.

In the second case, we assume 2x-1 is negative:
-(2x-1) = 5
This simplifies to -2x + 1 = 5
Solving for x, we get x = -2.

So in this example, we have both a positive solution (x = 3) and a negative solution (x = -2).

However, in some cases, you may only end up with positive solutions or only negative solutions. For example, if you have the equation |11-2x| = 5, the two cases would be:

First case (assuming 11-2x is positive):
11-2x = 5
Solving for x, we get x = 3.

Second case (assuming 11-2x is negative):
-(11-2x) = 5
This simplifies to -11 + 2x = 5
Solving for x, we get x = 8.

In this case, both solutions are positive.

In conclusion, when solving equations involving absolute values, you may have both negative and positive solutions, or only positive solutions, or only negative solutions. It depends on the specific equation you are working with.