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Hope this is helpful! Best wishes!Answer: [B]: " x = - 1 " and: " x = - 5 ".
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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 = {−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 both the positive value of that expression and 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 " and: "x = 5" ; which is: Answer choice: [B}.
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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!