Answer :
To solve the equation [tex]\(4|x+5|=16\)[/tex], let's break down the steps involved:
1. Simplify the equation:
- Divide both sides by 4 to isolate the absolute value:
[tex]\[
|x + 5| = 4
\][/tex]
2. Consider the two cases for the absolute value:
- The expression [tex]\(|x+5|\)[/tex] means that [tex]\(x+5\)[/tex] can be either 4 or [tex]\(-4\)[/tex]. This is because the absolute value of a number is its distance from zero on a number line, and it can be obtained in two ways: one positive, one negative.
3. Case 1: [tex]\(x + 5 = 4\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
4. Case 2: [tex]\(x + 5 = -4\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
Thus, the solutions 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. Simplify the equation:
- Divide both sides by 4 to isolate the absolute value:
[tex]\[
|x + 5| = 4
\][/tex]
2. Consider the two cases for the absolute value:
- The expression [tex]\(|x+5|\)[/tex] means that [tex]\(x+5\)[/tex] can be either 4 or [tex]\(-4\)[/tex]. This is because the absolute value of a number is its distance from zero on a number line, and it can be obtained in two ways: one positive, one negative.
3. Case 1: [tex]\(x + 5 = 4\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
4. Case 2: [tex]\(x + 5 = -4\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
Thus, the solutions 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]