Question

|x-5| = | 1 + x|

Solve for x

Answer 1

To solve the equation |x - 5| = |1 + x|, you need to consider two cases: one for when the expression inside the absolute value bars is non-negative and one for when it's negative. The key is to find the values of x that satisfy both cases.

Case 1: (x - 5) is non-negative, and (1 + x) is non-negative.

When both expressions are non-negative, the absolute value signs can be removed:

x - 5 = 1 + x

Now, try to solve for x:

x - x = 1 + 5

0 = 6

The equation 0 = 6 is a contradiction, which means that there are no solutions in this case.

Case 2: (x - 5) is non-negative, and (1 + x) is negative.

When (x - 5) is non-negative, you can remove the absolute value from it, but you need to negate (1 + x) inside the other absolute value:

x - 5 = -(1 + x)

Now, solve for x:

x - 5 = -1 - x

Add x to both sides:

2x - 5 = -1

Add 5 to both sides:

2x = 4

Divide by 2:

x = 2

So, in this case, x = 2 is a solution.

Now, you need to check if x = 2 satisfies the original equation:

|x - 5| = |1 + x|

|2 - 5| = |1 + 2|

| -3 | = | 3 |

Both sides are equal to 3, so x = 2 is indeed a solution to the original equation.

In summary, the solution to the equation |x - 5| = |1 + x| is x = 2.

Answer 2

Explanation:

To solve the equation |x - 5| = |1 + x|, we need to consider different cases based on the signs of the expressions inside the absolute values.

1. When (x - 5) and (1 + x) are both positive:

x - 5 = 1 + x

In this case, x cancels out, and you get -5 = 1, which is not true. So, there are no solutions in this case.

2. When (x - 5) is positive and (1 + x) is negative:

x - 5 = -(1 + x)

Solve for x:

x - 5 = -1 - x

2x = 4

x = 2

3. When (x - 5) is negative and (1 + x) is positive:

-(x - 5) = 1 + x

Solve for x:

-x + 5 = 1 + x

2x = 4

x = 2

So, the solutions are x = 2 for the cases where (x - 5) is positive and (1 + x) is negative, or vice versa.