Verified Answer :

  • Analyze the expression \(4x + 4\) and test each provided option.
  • Simplify \(4(x + 4)\) to get \(4x + 16\), which is not equivalent.
  • Simplify \(2(2x + 2)\) to get \(4x + 4\), which is equivalent.
  • Combine like terms in \(x + 3x + 4\) and \(2x + 1 + 2x + 3\) to confirm they both result in \(4x + 4\), and are thus equivalent.
  • The equivalent expressions are \(\boxed{2(2x + 2), x + 3x + 4, 2x + 1 + 2x + 3}\).

Explanation

  1. Analysis of the Problem

To determine which expressions are equivalent to the expression \(4x + 4\), we analyze each of the four given options one by one by simplifying them using algebraic properties such as the distributive property and combining like terms. Two expressions are considered equivalent if they represent the same value for any value of \(x\).

  1. Checking the First Expression: \(4(x + 4)\)

First, let's look at the expression \(4(x + 4)\). We use the distributive property, which states that \(a(b + c) = ab + ac\). Applying this, we get:
\[4(x + 4) = 4 \times x + 4 \times 4 = 4x + 16\]
Since \(4x + 16 eq 4x + 4\), this expression is not equivalent.

  1. Checking the Second Expression: \(2(2x + 2)\)

Next, we examine the expression \(2(2x + 2)\). Applying the distributive property:
\[2(2x + 2) = 2 \times 2x + 2 \times 2 = 4x + 4\]
Since the result is identical to the original expression \(4x + 4\), this expression is equivalent.

  1. Checking the Third Expression: \(x + 3x + 4\)

Now, let's simplify the expression \(x + 3x + 4\). By combining like terms (the coefficients of \(x\)), we get:
\[x + 3x + 4 = (1 + 3)x + 4 = 4x + 4\]
Because this result is equal to \(4x + 4\), this expression is also equivalent.

  1. Checking the Fourth Expression: \(2x + 1 + 2x + 3\)

Finally, we look at the expression \(2x + 1 + 2x + 3\). To simplify this, we group the terms with \(x\) and the constant terms together:
\[2x + 1 + 2x + 3 = (2x + 2x) + (1 + 3) = 4x + 4\]
Since this matches the original expression, this expression is also equivalent.

  1. Conclusion

In conclusion, the expressions equivalent to \(4x + 4\) are:

  • \(2(2x + 2)\)
  • \(x + 3x + 4\)
  • \(2x + 1 + 2x + 3\)

These are the expressions that, after simplification, result in exactly \(4x + 4\). The final answer is \(\boxed{2(2x + 2), x + 3x + 4, 2x + 1 + 2x + 3}\).

Examples

In real-world budgeting, you might have different ways to express the total cost of a project. For instance, if you pay a fixed fee of \$4 and then \$4 per hour of work, your total cost can be written as \(4x + 4\). If you contract a service that charges \$2 per hour for a two-person team plus a \$4 service fee, the cost is \(2(2x + 2)\), which is mathematically the same as your initial estimate. Understanding these algebraic relationships helps in verifying that different payment plans or material lists represent the same total expenditure, ensuring your calculations remain consistent.

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