|x-2|=|4+x|
How can I find the value of X?

  x = -1

Step-by-step explanation:

You want to find the value(s) of x that satisfy the equation |x-2| = |4+x|.

The behavior of each absolute value function changes at the point where its argument is zero. For this equation, that will be two points:

  x -2 = 0   ⇒   x = 2

  x +4 = 0   ⇒   x = -4

These two points on the number line divide the domain of the equation into three parts:

The behavior of the equation will be different in those three domains.

In this domain, both absolute value functions negate their arguments. That means the equation is equivalent to ...

  -(x -2) = -(4 +x)

  2 = -4 . . . . . . . . . . add x to both sides

This has no solutions: there are no values of x that will make this true.

In this domain, the absolute value function on the left negates its argument, so the equation is equivalent to ...

  -(x -2) = (4 +x)

  -2 = 2x . . . . . . . . . add x-4 to both sides

  -1 = x . . . . . . . . . divide by 2

This value of x is in the domain, so represents a solution to the equation.

  x = -1

In this domain, neither absolute value function negates its argument, so the equation is equivalent to ...

  x -2 = 4 +x

  -2 = 4 . . . . . . . subtract x

This has no solutions: there are no values of x that will make this true.

The given equation has one solution: x = -1.

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Additional comment

When solving an equation like this graphically, it is often useful to put it in the form f(x) = 0. We can obtain that form by defining ...

  f(x) = |x -2| -|4 +x|

The graph of this is attached. The only solution is where f(x) = 0, at x = -1.

Note that the function is parallel to the x-axis outside of the middle domain, so will not have any x-intercepts in those regions.

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step-by-step explanation:

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step-by-step explanation: