Select the correct answer.
Which statement is true about this equation?
[tex]$-9(x+3)+12=-3(2 x+5)-3 x$[/tex]
A. The equation has one solution, [tex]$x=1$[/tex].
B. The equation has one solution, [tex]$x=0$[/tex].
C. The equation has no solution.
D. The equation has infinitely many solutions.
Answer (2)
Expand both sides of the equation: − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x becomes − 9 x − 15 = − 6 x − 15 − 3 x .
Simplify both sides: − 9 x − 15 = − 9 x − 15 .
Add 9 x to both sides: − 15 = − 15 .
Since the equation simplifies to a true statement, the equation has infinitely many solutions. D
Explanation
Understanding the Problem We are given the equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x and asked to determine the nature of its solution(s). This means we need to find out if the equation has one solution, no solution, or infinitely many solutions.
Expanding Both Sides First, we expand both sides of the equation: − 9 ( x + 3 ) + 12 = − 9 x − 27 + 12 = − 9 x − 15 − 3 ( 2 x + 5 ) − 3 x = − 6 x − 15 − 3 x = − 9 x − 15
Simplified Equation Now we have the equation − 9 x − 15 = − 9 x − 15 .
Adding 9x to Both Sides Next, we add 9 x to both sides of the equation: − 9 x − 15 + 9 x = − 9 x − 15 + 9 x
− 15 = − 15
Conclusion Since the equation simplifies to − 15 = − 15 , which is a true statement, the equation has infinitely many solutions. This means that any value of x will satisfy the original equation.
Examples
Understanding equations and their solutions is crucial in many real-world applications. For instance, consider a scenario where you're comparing two different phone plans. Each plan has a monthly fee and a per-minute charge. If the equation representing the total cost of both plans simplifies to a true statement like -15 = -15, it means the plans cost the same regardless of the number of minutes you use. This helps you make an informed decision based on other factors, such as data allowance or network coverage.
The equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x simplifies to a true statement, indicating it has infinitely many solutions. Thus, the correct answer is option D. This means any value of x satisfies the equation.
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