What is an equivalent expression for 1.5(3x+4)+0.25(6x+8)?
The Correct Answer and Explanation is:
To find an equivalent expression for (1.5(3x + 4) + 0.25(6x + 8)), we can start by distributing each term in the expression.
Step 1: Distribute the constants
First, we distribute (1.5) to both terms inside the first parentheses:
[
1.5(3x) + 1.5(4) = 4.5x + 6
]
Next, we distribute (0.25) to both terms inside the second parentheses:
[
0.25(6x) + 0.25(8) = 1.5x + 2
]
Step 2: Combine the results
Now, we can combine both distributed results:
[
1.5(3x + 4) + 0.25(6x + 8) = (4.5x + 6) + (1.5x + 2)
]
Step 3: Combine like terms
Next, we combine the like terms (4.5x) and (1.5x), as well as the constants (6) and (2):
- For the (x) terms:
[
4.5x + 1.5x = 6x
] - For the constant terms:
[
6 + 2 = 8
]
Final Step: Write the equivalent expression
Combining these results gives us:
[
6x + 8
]
Conclusion
Thus, the equivalent expression for (1.5(3x + 4) + 0.25(6x + 8)) is:
[
\boxed{6x + 8}
]
Explanation
This problem illustrates the process of simplifying expressions through distribution and combining like terms. The distributive property is crucial for breaking down expressions where a term multiplies a sum. By applying it systematically, we can simplify complex expressions into simpler forms. In this case, we started with a compound expression involving both multiplication and addition, making it essential to handle each part step-by-step. The final result, (6x + 8), not only represents the original expression but also highlights how mathematical operations can be manipulated for clarity and simplicity. This technique is foundational in algebra, allowing for the easy manipulation and solving of equations in subsequent problems.