Answer :
Let's solve the equation [tex]\( |x + 5| - 6 = 1 \)[/tex].
### Step 1: Isolate the Absolute Value
First, we add 6 to both sides of the equation:
[tex]\[ |x + 5| = 7 \][/tex]
### Step 2: Consider the Definition of Absolute Value
The equation [tex]\( |x + 5| = 7 \)[/tex] means that the expression inside the absolute value, [tex]\( x + 5 \)[/tex], can be either 7 or -7. This gives us two separate equations to solve:
1. [tex]\( x + 5 = 7 \)[/tex]
2. [tex]\( x + 5 = -7 \)[/tex]
### Step 3: Solve Each Equation
Equation 1:
[tex]\[ x + 5 = 7 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = 7 - 5 \][/tex]
[tex]\[ x = 2 \][/tex]
Equation 2:
[tex]\[ x + 5 = -7 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = -7 - 5 \][/tex]
[tex]\[ x = -12 \][/tex]
### Step 4: Verify the Solutions
Let's verify these solutions by plugging them back into the original equation to ensure they satisfy [tex]\( |x + 5| - 6 = 1 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
- Inside the absolute value, [tex]\( x + 5 = 2 + 5 = 7 \)[/tex]
- Therefore, [tex]\( |x + 5| = |7| = 7 \)[/tex]
- Substitute back: [tex]\( 7 - 6 = 1 \)[/tex] - This is true.
For [tex]\( x = -12 \)[/tex]:
- Inside the absolute value, [tex]\( x + 5 = -12 + 5 = -7 \)[/tex]
- Therefore, [tex]\( |x + 5| = |-7| = 7 \)[/tex]
- Substitute back: [tex]\( 7 - 6 = 1 \)[/tex] - This is true.
Both solutions satisfy the original equation.
### Conclusion
The solution to the equation [tex]\( |x+5|-6=1 \)[/tex] is [tex]\( x = 2 \)[/tex] and [tex]\( x = -12 \)[/tex].
The answer is not explicitly listed among the given choices, which indicates a possible error in the multiple-choice options. If no listed option includes both [tex]\(-12\)[/tex] and [tex]\(2\)[/tex], the correct answer may be missing. Always double-check or consult with your instructor if in doubt!
### Step 1: Isolate the Absolute Value
First, we add 6 to both sides of the equation:
[tex]\[ |x + 5| = 7 \][/tex]
### Step 2: Consider the Definition of Absolute Value
The equation [tex]\( |x + 5| = 7 \)[/tex] means that the expression inside the absolute value, [tex]\( x + 5 \)[/tex], can be either 7 or -7. This gives us two separate equations to solve:
1. [tex]\( x + 5 = 7 \)[/tex]
2. [tex]\( x + 5 = -7 \)[/tex]
### Step 3: Solve Each Equation
Equation 1:
[tex]\[ x + 5 = 7 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = 7 - 5 \][/tex]
[tex]\[ x = 2 \][/tex]
Equation 2:
[tex]\[ x + 5 = -7 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = -7 - 5 \][/tex]
[tex]\[ x = -12 \][/tex]
### Step 4: Verify the Solutions
Let's verify these solutions by plugging them back into the original equation to ensure they satisfy [tex]\( |x + 5| - 6 = 1 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
- Inside the absolute value, [tex]\( x + 5 = 2 + 5 = 7 \)[/tex]
- Therefore, [tex]\( |x + 5| = |7| = 7 \)[/tex]
- Substitute back: [tex]\( 7 - 6 = 1 \)[/tex] - This is true.
For [tex]\( x = -12 \)[/tex]:
- Inside the absolute value, [tex]\( x + 5 = -12 + 5 = -7 \)[/tex]
- Therefore, [tex]\( |x + 5| = |-7| = 7 \)[/tex]
- Substitute back: [tex]\( 7 - 6 = 1 \)[/tex] - This is true.
Both solutions satisfy the original equation.
### Conclusion
The solution to the equation [tex]\( |x+5|-6=1 \)[/tex] is [tex]\( x = 2 \)[/tex] and [tex]\( x = -12 \)[/tex].
The answer is not explicitly listed among the given choices, which indicates a possible error in the multiple-choice options. If no listed option includes both [tex]\(-12\)[/tex] and [tex]\(2\)[/tex], the correct answer may be missing. Always double-check or consult with your instructor if in doubt!