Properties of Real numbers worksheet. - Ms. Coxs website - Free Printable

Problem: Solve the given problems related to properties of real numbers and explain each solution.

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#### Problem 1: Which expression must be added to \(3x - 7\) to equal \(0\)?

We need to find an expression \(E\) such that:
\[
(3x - 7) + E = 0
\]
To isolate \(E\), we subtract \(3x - 7\) from both sides:
\[
E = 0 - (3x - 7)
\]
Simplify the right-hand side:
\[
E = -3x + 7
\]
Thus, the expression that must be added is:
\[
\boxed{4}
\]

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#### 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:
\[
\frac{4}{3}
\]
because:
\[
\frac{3}{4} \cdot \frac{4}{3} = 1
\]
Thus, the multiplicative inverse is:
\[
\boxed{4}
\]

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#### Problem 3: The multiplicative inverse of \(-\frac{1}{3}\) is

The multiplicative inverse of a number \(a\) is a number \(b\) such that:
\[
a \cdot b = 1
\]
For \(-\frac{1}{3}\), the multiplicative inverse is:
\[
-3
\]
because:
\[
-\frac{1}{3} \cdot (-3) = 1
\]
Thus, the multiplicative inverse is:
\[
\boxed{3}
\]

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#### 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:
\[
-\frac{1}{x}
\]
because:
\[
\frac{1}{x} + \left(-\frac{1}{x}\right) = 0
\]
Thus, the additive inverse is:
\[
\boxed{2}
\]

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#### Problem 5: The reciprocal of 5 is

The reciprocal of a number \(a\) is a number \(b\) such that:
\[
a \cdot b = 1
\]
For \(5\), the reciprocal is:
\[
\frac{1}{5}
\]
because:
\[
5 \cdot \frac{1}{5} = 1
\]
Thus, the reciprocal is:
\[
\boxed{2}
\]

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#### 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:
\[
-\frac{2}{3}
\]
because:
\[
\frac{2}{3} + \left(-\frac{2}{3}\right) = 0
\]
Thus, the additive inverse is:
\[
\boxed{1}
\]

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#### 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:
\[
-(a - b) = -a + b
\]
Thus, the additive inverse is:
\[
\boxed{3}
\]

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#### Problem 8: If \(a \neq 0\) and the sum of \(x\) and \(\frac{1}{a}\) is 0, then

We are given:
\[
x + \frac{1}{a} = 0
\]
To solve for \(x\), subtract \(\frac{1}{a}\) from both sides:
\[
x = -\frac{1}{a}
\]
Thus, the correct answer is:
\[
\boxed{3}
\]

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#### Problem 9: The reciprocal of the expression \(\frac{2}{x} + \frac{3}{1}\) is

First, simplify the expression \(\frac{2}{x} + \frac{3}{1}\):
\[
\frac{2}{x} + 3 = \frac{2}{x} + \frac{3x}{x} = \frac{2 + 3x}{x}
\]
The reciprocal of \(\frac{2 + 3x}{x}\) is:
\[
\frac{x}{2 + 3x}
\]
Thus, the reciprocal is:
\[
\boxed{2}
\]

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#### 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. \(1 - x = x\)
2. \(x \cdot \frac{1}{x} = 1\)
3. \(1 \cdot 0 = 0\)
4. \(-1 \cdot x = -x\)

The equation that illustrates the multiplicative inverse property is:
\[
x \cdot \frac{1}{x} = 1
\]
Thus, the correct answer is:
\[
\boxed{2}
\]

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Final Answers:

1. \(\boxed{4}\)
2. \(\boxed{4}\)
3. \(\boxed{3}\)
4. \(\boxed{2}\)
5. \(\boxed{2}\)
6. \(\boxed{1}\)
7. \(\boxed{3}\)
8. \(\boxed{3}\)
9. \(\boxed{2}\)
10. \(\boxed{2}\)

Boxed Final Answer:
\[
\boxed{4, 4, 3, 2, 2, 1, 3, 3, 2, 2}
\]