Math worksheet reviewing properties of real numbers with multiple-choice questions.

Problem: Solve the given math problems related to properties of real numbers.

#### Problem 1: Which expression must be added to \(3x - 7\) to equal \(0\)?
- We need to find an expression \(E\) such that:
\[
(3x - 7) + E = 0
\]
- Solving for \(E\):
\[
E = 0 - (3x - 7) = -3x + 7
\]
- Therefore, the correct answer is:
\[
\boxed{4}
\]

#### Problem 2: What is the multiplicative inverse of \(\frac{3}{4}\)?
- The multiplicative inverse of a number \(a\) is a number \(b\) such that:
\[
a \cdot b = 1
\]
- For \(\frac{3}{4}\), the multiplicative inverse is:
\[
\frac{4}{3}
\]
- Therefore, the correct answer is:
\[
\boxed{4}
\]

#### Problem 3: The multiplicative inverse of \(-\frac{1}{3}\) is
- The multiplicative inverse of \(-\frac{1}{3}\) is a number \(b\) such that:
\[
-\frac{1}{3} \cdot b = 1
\]
- Solving for \(b\):
\[
b = -3
\]
- Therefore, the correct answer is:
\[
\boxed{4}
\]

#### Problem 4: The additive inverse of \(\frac{1}{x}\) is
- The additive inverse of a number \(a\) is a number \(b\) such that:
\[
a + b = 0
\]
- For \(\frac{1}{x}\), the additive inverse is:
\[
-\frac{1}{x}
\]
- Therefore, the correct answer is:
\[
\boxed{2}
\]

#### Problem 5: The reciprocal of 5 is
- The reciprocal of a number \(a\) is \(\frac{1}{a}\).
- For 5, the reciprocal is:
\[
\frac{1}{5}
\]
- Therefore, the correct answer is:
\[
\boxed{2}
\]

#### Problem 6: What is the additive inverse of \(\frac{2}{3}\)?
- The additive inverse of a number \(a\) is a number \(b\) such that:
\[
a + b = 0
\]
- For \(\frac{2}{3}\), the additive inverse is:
\[
-\frac{2}{3}
\]
- Therefore, the correct answer is:
\[
\boxed{1}
\]

#### Problem 7: What is the additive inverse of the expression \(a - b\)?
- The additive inverse of a number \(a\) is a number \(b\) such that:
\[
a + b = 0
\]
- For the expression \(a - b\), the additive inverse is:
\[
-(a - b) = -a + b
\]
- Therefore, the correct answer is:
\[
\boxed{3}
\]

#### Problem 8: If \(a \neq 0\) and the sum of \(x\) and \(\frac{1}{a}\) is 0, then
- Given:
\[
x + \frac{1}{a} = 0
\]
- Solving for \(x\):
\[
x = -\frac{1}{a}
\]
- Therefore, the correct answer is:
\[
\boxed{3}
\]

#### Problem 9: The reciprocal of the expression \(\frac{2}{x} + \frac{3}{1}\) is
- First, simplify the expression:
\[
\frac{2}{x} + \frac{3}{1} = \frac{2}{x} + 3 = \frac{2 + 3x}{x}
\]
- The reciprocal of \(\frac{2 + 3x}{x}\) is:
\[
\frac{x}{2 + 3x}
\]
- Therefore, the correct answer is:
\[
\boxed{2}
\]

#### Problem 10: Which equation illustrates the multiplicative inverse property?
- The multiplicative inverse property states that for any nonzero number \(a\), there exists a number \(b\) such that:
\[
a \cdot b = 1
\]
- Among the given options:
1. \(1 \cdot x = x\) (This is the multiplicative identity property.)
2. \(x \cdot \frac{1}{x} = 1\) (This illustrates the multiplicative inverse property.)
3. \(1 \cdot 0 = 0\) (This is the multiplicative property of zero.)
4. \(-1 \cdot x = -x\) (This is the property of multiplying by \(-1\).)
- Therefore, the correct answer is:
\[
\boxed{2}
\]

Final Answers:

\[
\boxed{4, 4, 4, 2, 2, 1, 3, 3, 2, 2}
\]