Answer :
Sure! Let's solve the equation [tex]\(4|x+5|=24\)[/tex] step by step.
1. Isolate the Absolute Value:
Start by dividing both sides of the equation by 4 to make it simpler:
[tex]\[
|x + 5| = \frac{24}{4} = 6
\][/tex]
2. Consider the Two Cases of the Absolute Value Equation:
An absolute value equation [tex]\( |x + 5| = 6 \)[/tex] means that the expression inside the absolute value can either be positive or negative. Therefore, we can split it into two separate equations:
- Case 1: [tex]\( x + 5 = 6 \)[/tex]
- Case 2: [tex]\( x + 5 = -6 \)[/tex]
3. Solve Each Case:
- For Case 1: [tex]\( x + 5 = 6 \)[/tex]
[tex]\[
x = 6 - 5 = 1
\][/tex]
- For Case 2: [tex]\( x + 5 = -6 \)[/tex]
[tex]\[
x = -6 - 5 = -11
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\(4|x+5|=24\)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -11 \)[/tex].
Therefore, the correct answer is D. [tex]\( x = -11 \)[/tex] and [tex]\( x = 1 \)[/tex].
1. Isolate the Absolute Value:
Start by dividing both sides of the equation by 4 to make it simpler:
[tex]\[
|x + 5| = \frac{24}{4} = 6
\][/tex]
2. Consider the Two Cases of the Absolute Value Equation:
An absolute value equation [tex]\( |x + 5| = 6 \)[/tex] means that the expression inside the absolute value can either be positive or negative. Therefore, we can split it into two separate equations:
- Case 1: [tex]\( x + 5 = 6 \)[/tex]
- Case 2: [tex]\( x + 5 = -6 \)[/tex]
3. Solve Each Case:
- For Case 1: [tex]\( x + 5 = 6 \)[/tex]
[tex]\[
x = 6 - 5 = 1
\][/tex]
- For Case 2: [tex]\( x + 5 = -6 \)[/tex]
[tex]\[
x = -6 - 5 = -11
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\(4|x+5|=24\)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -11 \)[/tex].
Therefore, the correct answer is D. [tex]\( x = -11 \)[/tex] and [tex]\( x = 1 \)[/tex].