_____
Hope this is helpful! Best wishes!Answer: [B]: " x = - 1 " <u>and</u>: " x = - 5 ".
_____
We are given the following equation:
−3|2x + 6| = −12 ; Solve for "x" ; & we are given answer choices from which to choose.
→ So, let us examine the equation.
→On the "left hand side" of the equation; we are given: "-3" ;
→ multiplied by {the 'absolute value' of [the expression: "2x + 6"] };
→ followed by an "equals sign" ; then followed by the "right hand side" of the equation—which is the number: " -12 ".
To get rid of the the "-3" on the "left-hand side" of the equation; we can divide Each Side of the equation by "-3" ; since on the "left-hand side" of the question: "-3/-3" = 1 ; and: "1" ; multiplied by the "absolute value expression" ; (on the "left-hand side of the equation"} is equal to:
that same "absolute value expression" ;
→ {since: "1" ; multiplied by "any value" ; results in that "exact same original value". Note that refers to the "identity property of multiplication."}.
On the "right hand side" of the equation: "-12/-3 = 12/ 3 = 4 " ;
So: Given the equation: " −3|2x + 6| = −12 " ; → that is: -3 *| 2x + 6 | = −12 ; Divide each side by: "-3" ; → { -3|2x + 6| } / -3 = {−12} / -3 ;
to get: " |2x + 6| = 4 " ; Now, let's solve for "x" in this "absolute value" equation:
Note that on the "left-hand side" :
→The expression within the 2 (two) "absolute value symbols must be equal to <u>both</u> the positive value of that expression <u>and</u> the negative value of that expression. As such, we shall solve for the values for "x" using "Case 1" and "Case 2" scenarios:
Case 1) 2x + 6 = 4 ; Subtract "6" from each side of the equation:
→ 2x + 6 - 6 = 4 - 6 ; to get: 2x = -2 ;
Now divide each side of the equation by "-2" ;
to isolate "x" on one side of the equation; & to solve for "x" :
→ 2x/2 = -2/2 ; to get: " x = - 1 " .
Case 2) We have |2x + 6| = - 4
We shall solve for the "negative value" of the expression within the "absolute value" bars on the "left-hand side of the equation:
→ Write as: -(2x + 6) = 4 ; Solve for "x" ;
Rewrite this as: " -1(2x + 6) = 4 " ;
→ See explanation above about the "identity property of multiplication."
Method 1)
Divide each side of the equation by "-1" ; to get rid of the "-1" on the "left-hand side" of the equation<.
→ { -1(2x + 6) } / -1 = {4} / -1 " ;
to get: 2x + 6 = - 4 ; Now, we subtract "6" from each side of the equation:
→ 2x + 6 - 6 = - 4 - 6 ; to get: 2x = - 10 ; Now divide each side of the equation by "2" ; to isolate "x" on one side of the equation; & to solve for "x" :
→ 2x/2 = -10/2 ; to get: " x = - 5 " .
Method 2)
→ Write as: -(2x + 6) = 4 ; Solve for "x" ; Rewrite this as: " -1(2x + 6) = 4 " ;
→ See explanation above about the "identity property of multiplication."
Note the "distributive property" of multiplication:
→ a(b + c) = ab + ac ;
As such: On the "left-hand side of the equation:
→ "-1(2x + 6) = ( -1*2x) + (-1*6) = (-2x) + (-6) = -2x - 6 ;
So, rewrite the equation; & bring down the "4" on the "right-hand side":
→ "-2x - 6 = 4 " ; Now, we add "6" to each side of the equation:
→ -2x - 6 + 6 = 4 + 6 ; to get: -2x = 10 ; Now divide each side of the equation by "-2" ; to isolate "x" on one side of the equation; & to solve for "x" ; -2x /-2 = 10/-2; to get: "x = -5 " .
So; we have: "x = -1 " <u>and</u>: "x = 5" ; which is: Answer choice: [B}.
_____
Now, let us check both values of "x" by plugging them into our original given equation:
Given: −3 |2x + 6| = −12 ;
Start by substituting one of our solved values: " x = -1 " ; & see if the equation holds true:
→ -3 | (2(-1) + 6 | =? -12 ? ;
→ -3 |-2 + 6| =? -12 ? ;
→ -3 * |4| =? = -12 ? ;
→ -3 * 4 =? = - 12? Yes!
{Note: The {absolute value of "4" = 4.}. The absolute value of a quantity is magnitude of the both the positive and negative value of the quantity. The absolute of [a quantity] is represented by the enclosure of two (2) straight, vertical, slightly large line segments—; that is:
|(insert number or other quantity)| ;
The absolute value of "-4" is "4" ; that is: |-4| = 4 ; and the absolute value of "4" is "4" ; that is: |4| = 4. The absolute value of "0" [zero] is "0" [zero]; that is: |0| = 0. For numbers greater that "0"; the absolute value of a number is that number. For numbers smaller than "0" [i.e. negative numbers]; the absolute value would be the positive value of that number.
Now, let us check our work further; by substituting our other "solved value" for "x" ; that is: " x = -5 " ; into the original equation; & see if the equation holds true:
Given: " −3|2x + 6| = − 12 "; Plug in "-5" for "x" ;
→ -3 | (2(-5) + 6 | =? -12 ? ;
→ -3 | -10 + 6 | =? -12 ? ;
→ -3 * |-4| =? -12 ? '
Note: As mentioned above, the absolute value for "-4" is "4" ;
→ that is: |-4 | = 4 ;
As such: -3 * 4 =? -12 ? Yes!
→ So; BOTH of our 'obtained values' for "x" make sense!
Hope this is helpful!
Best wishes!
