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

Let's go through each question one by one and solve them step-by-step, explaining the reasoning.

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1. Which expression must be added to $3x - 7$ to equal 0?

We want:
$$
(3x - 7) + ? = 0
$$
So, we need the additive inverse of $3x - 7$, which is:
$$
-(3x - 7) = -3x + 7
$$

✔ Answer: (4) $-3x + 7$

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2. What is the multiplicative inverse of $\frac{3}{4}$?

The multiplicative inverse (or reciprocal) of a number is what you multiply it by to get 1.

$$
\frac{3}{4} \times ? = 1 \Rightarrow ? = \frac{4}{3}
$$

✔ Answer: (2) $\frac{4}{3}$

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3. The multiplicative inverse of $-\frac{1}{3}$ is

Again, find the reciprocal:
$$
-\frac{1}{3} \times ? = 1 \Rightarrow ? = -3
$$

Because:
$$
-\frac{1}{3} \times (-3) = 1
$$

✔ Answer: (4) $-3$

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4. The additive inverse of $\frac{1}{a}$ is

Additive inverse means the number that adds to zero:
$$
\frac{1}{a} + ? = 0 \Rightarrow ? = -\frac{1}{a}
$$

✔ Answer: (2) $-\frac{1}{a}$

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5. The reciprocal of 5 is

Reciprocal of 5 is:
$$
\frac{1}{5}
$$

✔ Answer: (2) $\frac{1}{5}$

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6. What is the additive inverse of $\frac{2}{3}$?

Additive inverse of $\frac{2}{3}$ is:
$$
-\frac{2}{3}
$$

✔ Answer: (3) $-\frac{2}{3}$

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7. What is the additive inverse of the expression $a - b$?

Additive inverse of $a - b$ is:
$$
-(a - b) = -a + b = b - a
$$

But looking at choices:
- (1) $a + b$
- (2) $a - b$
- (3) $-a + b$ → this matches!
- (4) $-a - b$

So, $-a + b$ is correct.

✔ Answer: (3) $-a + b$

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8. If $a \neq 0$ and the sum of $x$ and $\frac{1}{a}$ is 0, then...

We are told:
$$
x + \frac{1}{a} = 0 \Rightarrow x = -\frac{1}{a}
$$

✔ Answer: (3) $x = -\frac{1}{a}$

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9. The reciprocal of the expression $2 + \frac{3}{x}$ is

First, simplify the expression:
$$
2 + \frac{3}{x} = \frac{2x}{x} + \frac{3}{x} = \frac{2x + 3}{x}
$$

Now, take the reciprocal:
$$
\text{Reciprocal} = \frac{x}{2x + 3}
$$

✔ Answer: (2) $\frac{x}{2x + 3}$

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10. Which equation illustrates the multiplicative inverse property?

The multiplicative inverse property states:
$$
a \cdot \frac{1}{a} = 1 \quad \text{(for } a \ne 0\text{)}
$$

Let’s examine the options:

1) $1 \cdot x = x$ → This is the identity property, not inverse.
2) $x \cdot \frac{1}{x} = 1$ → ✔ Yes! This is the multiplicative inverse.
3) $1 \cdot 0 = 0$ → Multiplicative identity with zero.
4) $-1 \cdot x = -x$ → Distributive or negative multiplication.

✔ Answer: (2) $x \cdot \frac{1}{x} = 1$

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✔ Final Answers Summary:

| Question | Answer |
|--------|--------|
| 1 | (4) $-3x + 7$ |
| 2 | (2) $\frac{4}{3}$ |
| 3 | (4) $-3$ |
| 4 | (2) $-\frac{1}{a}$ |
| 5 | (2) $\frac{1}{5}$ |
| 6 | (3) $-\frac{2}{3}$ |
| 7 | (3) $-a + b$ |
| 8 | (3) $x = -\frac{1}{a}$ |
| 9 | (2) $\frac{x}{2x + 3}$ |
| 10 | (2) $x \cdot \frac{1}{x} = 1$ |

Let me know if you'd like this in a printable format or explained further!