Question
Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2(4−8x)+5x=1−7x
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Answers
Answered by
GPT-5 mini
Solve 3/2(4 − 8x) + 5x = 1 − 7x with justifications.
1. Apply the distributive property: (3/2)·4 − (3/2)·8x + 5x = 1 − 7x → 6 − 12x + 5x = 1 − 7x.
(Because (3/2)·4 = 6 and (3/2)·8x = 12x.)
2. Combine like terms on the left (add the x-terms): 6 − 7x = 1 − 7x.
(Using the associative and commutative properties of addition.)
3. Add 7x to both sides to eliminate the x-term (addition property of equality): 6 = 1.
4. 6 = 1 is a contradiction (false statement), so no value of x satisfies the equation.
Conclusion: The equation has 0 solutions; the solution set is empty (the equation is inconsistent).
1. Apply the distributive property: (3/2)·4 − (3/2)·8x + 5x = 1 − 7x → 6 − 12x + 5x = 1 − 7x.
(Because (3/2)·4 = 6 and (3/2)·8x = 12x.)
2. Combine like terms on the left (add the x-terms): 6 − 7x = 1 − 7x.
(Using the associative and commutative properties of addition.)
3. Add 7x to both sides to eliminate the x-term (addition property of equality): 6 = 1.
4. 6 = 1 is a contradiction (false statement), so no value of x satisfies the equation.
Conclusion: The equation has 0 solutions; the solution set is empty (the equation is inconsistent).
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