Answer :
To solve the equation [tex]\( 4|x+6| = 20 \)[/tex], let's follow these steps:
1. Isolate the absolute value term:
[tex]\[ 4|x+6| = 20 \][/tex]
2. Divide both sides by 4 to simplify:
[tex]\[ |x+6| = \frac{20}{4} \implies |x+6| = 5 \][/tex]
3. Solve the absolute value equation [tex]\( |x+6| = 5 \)[/tex] by considering the definition of absolute value:
This equation implies two cases:
Case 1: [tex]\( x + 6 = 5 \)[/tex]
[tex]\[ x + 6 = 5 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = 5 - 6 \implies x = -1 \][/tex]
Case 2: [tex]\( x + 6 = -5 \)[/tex]
[tex]\[ x + 6 = -5 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = -5 - 6 \implies x = -11 \][/tex]
Thus, the solutions to the equation [tex]\( 4|x+6| = 20 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].
From the given options, the correct answer is:
[tex]\[ \boxed{\text{A. } x = -11 \text{ and } x = -1} \][/tex]
1. Isolate the absolute value term:
[tex]\[ 4|x+6| = 20 \][/tex]
2. Divide both sides by 4 to simplify:
[tex]\[ |x+6| = \frac{20}{4} \implies |x+6| = 5 \][/tex]
3. Solve the absolute value equation [tex]\( |x+6| = 5 \)[/tex] by considering the definition of absolute value:
This equation implies two cases:
Case 1: [tex]\( x + 6 = 5 \)[/tex]
[tex]\[ x + 6 = 5 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = 5 - 6 \implies x = -1 \][/tex]
Case 2: [tex]\( x + 6 = -5 \)[/tex]
[tex]\[ x + 6 = -5 \][/tex]
Subtract 6 from both sides:
[tex]\[ x = -5 - 6 \implies x = -11 \][/tex]
Thus, the solutions to the equation [tex]\( 4|x+6| = 20 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].
From the given options, the correct answer is:
[tex]\[ \boxed{\text{A. } x = -11 \text{ and } x = -1} \][/tex]
