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 = − 9 x − 15 .
Simplify the equation: Adding 9 x to both sides results in − 15 = − 15 .
Since the equation simplifies to a true statement ( − 15 = − 15 ), the equation has infinitely many solutions.
Therefore, the correct answer is D: The equation has infinitely many solutions. D
Explanation
Analyzing the equation We are given the equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x and we need to determine which statement is true about it. The options are: A. The equation has one solution, x = 1 .
B. The equation has one solution, x = 0 .
C. The equation has no solution. D. The equation has 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: − 9 x − 15 = − 9 x − 15
Adding 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, the equation has infinitely many solutions. This means that any value of x will satisfy the equation.
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 has infinitely many solutions, it might indicate a design redundancy or a need for additional constraints to ensure stability and safety. Understanding the nature of solutions helps engineers refine their designs and avoid potential issues.
The equation − 9 ( x + 3 ) + 12 = − 3 ( 2 x + 5 ) − 3 x simplifies to − 15 = − 15 , which is always true. This means the equation has infinitely many solutions. The correct answer is D.
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