Answer :
Sure! Let's solve the equation [tex]\(4|x+5|=16\)[/tex] step-by-step.
### Step 1: Isolate the Absolute Value
First, we want to isolate the absolute value by dividing both sides of the equation by 4:
[tex]\[
|x+5| = \frac{16}{4}
\][/tex]
[tex]\[
|x+5| = 4
\][/tex]
### Step 2: Solve the Absolute Value Equation
The equation [tex]\(|x+5| = 4\)[/tex] means that the expression inside the absolute value, [tex]\(x+5\)[/tex], can be either 4 or -4. This is because the absolute value of a number is its distance from zero on the number line, which makes it positive.
So we have two separate equations to solve:
1. [tex]\(x + 5 = 4\)[/tex]
2. [tex]\(x + 5 = -4\)[/tex]
### Step 3: Solve Each Equation
Equation 1: [tex]\(x + 5 = 4\)[/tex]
- Subtract 5 from both sides:
[tex]\[
x = 4 - 5
\][/tex]
- Simplify:
[tex]\[
x = -1
\][/tex]
Equation 2: [tex]\(x + 5 = -4\)[/tex]
- Subtract 5 from both sides:
[tex]\[
x = -4 - 5
\][/tex]
- Simplify:
[tex]\[
x = -9
\][/tex]
### Step 4: Write the Solutions
The solutions to the equation [tex]\(4|x+5|=16\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
### Conclusion
Therefore, the correct choice is:
C. [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
### Step 1: Isolate the Absolute Value
First, we want to isolate the absolute value by dividing both sides of the equation by 4:
[tex]\[
|x+5| = \frac{16}{4}
\][/tex]
[tex]\[
|x+5| = 4
\][/tex]
### Step 2: Solve the Absolute Value Equation
The equation [tex]\(|x+5| = 4\)[/tex] means that the expression inside the absolute value, [tex]\(x+5\)[/tex], can be either 4 or -4. This is because the absolute value of a number is its distance from zero on the number line, which makes it positive.
So we have two separate equations to solve:
1. [tex]\(x + 5 = 4\)[/tex]
2. [tex]\(x + 5 = -4\)[/tex]
### Step 3: Solve Each Equation
Equation 1: [tex]\(x + 5 = 4\)[/tex]
- Subtract 5 from both sides:
[tex]\[
x = 4 - 5
\][/tex]
- Simplify:
[tex]\[
x = -1
\][/tex]
Equation 2: [tex]\(x + 5 = -4\)[/tex]
- Subtract 5 from both sides:
[tex]\[
x = -4 - 5
\][/tex]
- Simplify:
[tex]\[
x = -9
\][/tex]
### Step 4: Write the Solutions
The solutions to the equation [tex]\(4|x+5|=16\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
### Conclusion
Therefore, the correct choice is:
C. [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].