Answer :
To solve the equation [tex]\(4|x+6|=16\)[/tex], follow these steps:
1. Divide Both Sides by 4:
First, divide both sides of the equation by 4 to isolate the absolute value:
[tex]\[
|x+6| = \frac{16}{4} = 4
\][/tex]
2. Consider the Definition of Absolute Value:
The absolute value equation [tex]\(|x+6| = 4\)[/tex] means that the expression inside the absolute value, [tex]\(x+6\)[/tex], can either be equal to 4 or -4. This gives us two separate equations to solve.
3. Solve the First Equation:
For the first case, set [tex]\(x+6 = 4\)[/tex]:
[tex]\[
x + 6 = 4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 4 - 6 = -2
\][/tex]
4. Solve the Second Equation:
For the second case, set [tex]\(x+6 = -4\)[/tex]:
[tex]\[
x + 6 = -4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -4 - 6 = -10
\][/tex]
5. Conclusion:
The solutions for the equation [tex]\(4|x+6|=16\)[/tex] are [tex]\(x = -2\)[/tex] and [tex]\(x = -10\)[/tex].
Therefore, the correct answer is:
C. [tex]\(x=-2\)[/tex] and [tex]\(x=-10\)[/tex]
1. Divide Both Sides by 4:
First, divide both sides of the equation by 4 to isolate the absolute value:
[tex]\[
|x+6| = \frac{16}{4} = 4
\][/tex]
2. Consider the Definition of Absolute Value:
The absolute value equation [tex]\(|x+6| = 4\)[/tex] means that the expression inside the absolute value, [tex]\(x+6\)[/tex], can either be equal to 4 or -4. This gives us two separate equations to solve.
3. Solve the First Equation:
For the first case, set [tex]\(x+6 = 4\)[/tex]:
[tex]\[
x + 6 = 4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 4 - 6 = -2
\][/tex]
4. Solve the Second Equation:
For the second case, set [tex]\(x+6 = -4\)[/tex]:
[tex]\[
x + 6 = -4
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -4 - 6 = -10
\][/tex]
5. Conclusion:
The solutions for the equation [tex]\(4|x+6|=16\)[/tex] are [tex]\(x = -2\)[/tex] and [tex]\(x = -10\)[/tex].
Therefore, the correct answer is:
C. [tex]\(x=-2\)[/tex] and [tex]\(x=-10\)[/tex]