Solve [tex]\( |x-5|+7=15 \)[/tex].

A. [tex]\( x=-13 \)[/tex] and [tex]\( x=3 \)[/tex]

B. [tex]\( x=-13 \)[/tex] and [tex]\( x=-3 \)[/tex]

C. [tex]\( x=13 \)[/tex] and [tex]\( x=-13 \)[/tex]

D. [tex]\( x=13 \)[/tex] and [tex]\( x=-3 \)[/tex]



Answer :

To solve the equation [tex]\( |x-5| + 7 = 15 \)[/tex], let's break the problem into a few manageable steps.

### Step 1: Isolate the Absolute Value Expression
First, we need to isolate the absolute value expression [tex]\( |x - 5| \)[/tex]. To do this, we'll subtract 7 from both sides of the equation:

[tex]\[ |x - 5| + 7 - 7 = 15 - 7 \][/tex]

This simplifies to:

[tex]\[ |x - 5| = 8 \][/tex]

### Step 2: Solve the Absolute Value Equation
The absolute value equation [tex]\( |x - 5| = 8 \)[/tex] means that the expression inside the absolute value can be either [tex]\( 8 \)[/tex] or [tex]\( -8 \)[/tex]. So, we set up two separate equations:

[tex]\[ x - 5 = 8 \quad \text{or} \quad x - 5 = -8 \][/tex]

#### Solving the First Equation
[tex]\[ x - 5 = 8 \][/tex]

Add 5 to both sides:

[tex]\[ x = 8 + 5 \][/tex]

[tex]\[ x = 13 \][/tex]

#### Solving the Second Equation
[tex]\[ x - 5 = -8 \][/tex]

Add 5 to both sides:

[tex]\[ x = -8 + 5 \][/tex]

[tex]\[ x = -3 \][/tex]

### Step 3: Verify the Solutions
The potential solutions are [tex]\( x = 13 \)[/tex] and [tex]\( x = -3 \)[/tex]. To ensure these are correct, we substitute them back into the original equation:

#### For [tex]\( x = 13 \)[/tex]
[tex]\[ |13 - 5| + 7 = |8| + 7 = 8 + 7 = 15 \][/tex]

This holds true.

#### For [tex]\( x = -3 \)[/tex]
[tex]\[ |-3 - 5| + 7 = |-8| + 7 = 8 + 7 = 15 \][/tex]

This also holds true.

### Step 4: Conclusion
Both solutions satisfy the original equation. Therefore, the correct answers are:

[tex]\[ x = 13 \quad \text{and} \quad x = -3 \][/tex]

Looking at the given answer choices, the correct answer is:

[tex]\[ \boxed{D} \][/tex]