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 = − 9 x − 15 and − 3 ( 2 x + 5 ) − 3 x = − 9 x − 15 .
Simplify the equation to − 9 x − 15 = − 9 x − 15 .
Add 9 x to both sides, resulting in − 15 = − 15 .
Since the equation is always true, the equation has infinitely many solutions. D
Explanation
Analyze the problem We are given the equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x and asked to determine its solution type. This means we need to find out if it has one solution, no solution, or infinitely many solutions. To do this, we will simplify both sides of the equation and see what we get.
Expand 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
Rewrite the equation Now we have − 9 x − 15 = − 9 x − 15 .
Add 9x to both sides Add 9 x to both sides: − 9 x − 15 + 9 x = − 9 x − 15 + 9 x
− 15 = − 15
Conclusion Since − 15 = − 15 is always true, regardless of the value of x , the equation has infinitely many solutions.
Examples
When designing a bridge, engineers use equations to model the forces and stresses acting on the structure. If an equation representing a certain aspect of the bridge's stability has infinitely many solutions, it indicates that the design is inherently stable under the modeled conditions, regardless of specific load distributions within the defined parameters. This ensures the bridge can withstand a range of loads without compromising its integrity.
The given equation simplifies to − 15 = − 15 , which is always true for any value of x . Therefore, the equation has infinitely many solutions, and the correct answer is option D.
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