Answer :
To solve the equation [tex]\(4|x+5|=16\)[/tex], let's work through it step by step.
1. Isolate the Absolute Value:
Start by dividing both sides of the equation by 4 to make the absolute value stand alone:
[tex]\[
|x+5| = \frac{16}{4} = 4
\][/tex]
2. Consider the Definition of Absolute Value:
The absolute value equation [tex]\( |x+5| = 4 \)[/tex] means that the expression inside the absolute value can be either 4 or -4. So, we set up two separate equations to solve:
- Case 1: [tex]\( x+5 = 4 \)[/tex]
- Case 2: [tex]\( x+5 = -4 \)[/tex]
3. Solve Each Equation:
Case 1:
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = 4 - 5 = -1
\][/tex]
Case 2:
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -4 - 5 = -9
\][/tex]
4. Solution:
The solutions to the equation are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].
Therefore, the correct answer is C. [tex]\(x=-1\)[/tex] and [tex]\(x=-9\)[/tex].
1. Isolate the Absolute Value:
Start by dividing both sides of the equation by 4 to make the absolute value stand alone:
[tex]\[
|x+5| = \frac{16}{4} = 4
\][/tex]
2. Consider the Definition of Absolute Value:
The absolute value equation [tex]\( |x+5| = 4 \)[/tex] means that the expression inside the absolute value can be either 4 or -4. So, we set up two separate equations to solve:
- Case 1: [tex]\( x+5 = 4 \)[/tex]
- Case 2: [tex]\( x+5 = -4 \)[/tex]
3. Solve Each Equation:
Case 1:
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = 4 - 5 = -1
\][/tex]
Case 2:
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -4 - 5 = -9
\][/tex]
4. Solution:
The solutions to the equation are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].
Therefore, the correct answer is C. [tex]\(x=-1\)[/tex] and [tex]\(x=-9\)[/tex].