Question

| 1 + | 3 + x || -2 = 0

Answer 1

$\displaystyle \bigg|1+|3+x|\bigg|-2=0\\\bigg|1+|3+x|\bigg|=2\\1+|3+x|=2; 1+|3+x|=-2\\\\\\ |3+x|=1; |3+x|=-3\\\\|3+x|=1; |3+x|\neq -3\\\\3+x=1; 3+x=-1\\\\x=-2;x=-4$

Answer 2

To solve the equation |1 + |3 + x|| - 2 = 0, you'll need to consider the absolute values.

Let's break it down step by step:

1. Start with the innermost absolute value: |3 + x|.

   • If 3 + x is positive, then |3 + x| = 3 + x.
   • If 3 + x is negative, then |3 + x| = -(3 + x).

2. Now, we have |1 + (3 + x)| - 2 = 0:

   • If 3 + x is positive, this becomes |1 + (3 + x)| - 2 = (1 + (3 + x)) - 2 = 4 + x - 2 = x + 2.
   • If 3 + x is negative, this becomes |1 + (3 + x)| - 2 = -(1 + (3 + x)) - 2 = -4 - x - 2 = -x - 6.

Now we have two possibilities:

1. If 3 + x is positive: x + 2 = 0

2. If 3 + x is negative: -x - 6 = 0

Let's solve each equation:

1. If 3 + x is positive: x + 2 = 0 Subtract 2 from both sides: x = -2

2. If 3 + x is negative: -x - 6 = 0 Add x to both sides: -6 = x

So, the solutions to the equation |1 + |3 + x|| - 2 = 0 are x = -2 and x = -6.

Answer 3

To solve the equation |1 + |3 + x|| - 2 = 0, you can break it down into cases based on the inner absolute value and the outer absolute value. Here's how you can do it step by step:

Case 1: 3 + x is non-negative (i.e., 3 + x ≥ 0)

In this case, the inner absolute value simplifies to:

|3 + x| = 3 + x

So the equation becomes:

|1 + (3 + x)| - 2 = 0

Simplify further:

|4 + x| - 2 = 0

Now, add 2 to both sides:

|4 + x| = 2

This gives us two possible equations:

1. 4 + x = 2
2. 4 + x = -2

Solve each equation separately:

1. 4 + x = 2 Subtract 4 from both sides: x = 2 - 4 x = -2

2. 4 + x = -2 Subtract 4 from both sides: x = -2 - 4 x = -6

So for this case, there are two solutions: x = -2 and x = -6.

Case 2: 3 + x is negative (i.e., 3 + x < 0)

In this case, the inner absolute value simplifies to:

|3 + x| = -(3 + x)

So the equation becomes:

|1 - (3 + x)| - 2 = 0

Simplify further:

|1 - 3 - x| - 2 = 0

|(-2 - x)| - 2 = 0

Now, add 2 to both sides:

|(-2 - x)| = 2

This gives us two possible equations:

1. -2 - x = 2
2. -2 - x = -2

Solve each equation separately:

1. -2 - x = 2 Add 2 to both sides: -x = 2 + 2 -x = 4 Divide by -1 (to isolate x): x = -4

2. -2 - x = -2 Add 2 to both sides: -x = -2 + 2 -x = 0 Divide by -1 (to isolate x): x = 0

So for this case, there are two more solutions: x = -4 and x = 0.

In summary, the equation |1 + |3 + x|| - 2 = 0 has four solutions: x = -2, x = -6, x = -4, and x = 0.