Question

use the drop downs to give the inverses, either additive or multiplicative, as necessary. (1 point)

1. the additive inverse of 4 = -4
2. the additive inverse of -9 = 9
3. the additive inverse of \\(\frac{2}{3}\\) = 2/3
4. the additive inverse of \\(-\frac{1}{12}\\) = -1/12

1. the multiplicative inverse of 4 = 1/4
2. the multiplicative inverse of -9 = -1/9
3. the multiplicative inverse of \\(\frac{2}{3}\\) = 2/3
4. the multiplicative inverse of \\(-\frac{1}{12}\\) = 1/12

Explanation:

Step1: Recall Additive Inverse

The additive inverse of a number \( a \) is \( -a \) because \( a + (-a) = 0 \).

• For \( 4 \), additive inverse is \( -4 \) (correct as \( 4 + (-4)=0 \)).
• For \( -9 \), additive inverse is \( 9 \) (correct as \( -9 + 9 = 0 \)).
• For \( \frac{2}{3} \), additive inverse should be \( -\frac{2}{3} \) (current answer \( \frac{2}{3} \) is wrong). Wait, no—wait, the problem's dropdown: maybe a typo? Wait, no, let's recheck. Wait, the user's problem shows the dropdowns, but let's correct the additive inverse of \( \frac{2}{3} \): additive inverse is \( -\frac{2}{3} \), and additive inverse of \( -\frac{1}{12} \) is \( \frac{1}{12} \) (current answer \( -\frac{1}{12} \) is wrong).

Step2: Recall Multiplicative Inverse

The multiplicative inverse of a number \( a \) (non - zero) is \( \frac{1}{a} \) because \( a\times\frac{1}{a}=1 \).

• For \( 4 \), multiplicative inverse is \( \frac{1}{4} \) (correct as \( 4\times\frac{1}{4}=1 \)).
• For \( -9 \), multiplicative inverse is \( -\frac{1}{9} \) (correct as \( -9\times(-\frac{1}{9}) = 1 \)).
• For \( \frac{2}{3} \), multiplicative inverse is \( \frac{3}{2} \) (current answer \( \frac{2}{3} \) is wrong).
• For \( -\frac{1}{12} \), multiplicative inverse is \( - 12 \) (current answer \( \frac{1}{12} \) is wrong).

Wait, but the problem is to check the dropdowns. Let's correct each:

Additive Inverses:

1. Additive inverse of \( 4 \): \( -4 \) (correct, since \( 4+(-4)=0 \)).
2. Additive inverse of \( -9 \): \( 9 \) (correct, since \( -9 + 9=0 \)).
3. Additive inverse of \( \frac{2}{3} \): Should be \( -\frac{2}{3} \) (current dropdown answer \( \frac{2}{3} \) is incorrect).
4. Additive inverse of \( -\frac{1}{12} \): Should be \( \frac{1}{12} \) (current dropdown answer \( -\frac{1}{12} \) is incorrect).

Multiplicative Inverses:

1. Multiplicative inverse of \( 4 \): \( \frac{1}{4} \) (correct, \( 4\times\frac{1}{4}=1 \)).
2. Multiplicative inverse of \( -9 \): \( -\frac{1}{9} \) (correct, \( -9\times(-\frac{1}{9}) = 1 \)).
3. Multiplicative inverse of \( \frac{2}{3} \): Should be \( \frac{3}{2} \) (current dropdown answer \( \frac{2}{3} \) is incorrect).
4. Multiplicative inverse of \( -\frac{1}{12} \): Should be \( - 12 \) (current dropdown answer \( \frac{1}{12} \) is incorrect).

But maybe the problem is to identify the correct dropdowns. Let's re - express the correct answers:

Additive Inverses:

1. The Additive Inverse of \( 4=-4 \) (correct as per the dropdown shown).
2. The Additive Inverse of \( -9 = 9 \) (correct as per the dropdown shown).
3. The Additive Inverse of \( \frac{2}{3}=-\frac{2}{3} \) (the dropdown shown has \( \frac{2}{3} \), which is wrong).
4. The Additive Inverse of \( -\frac{1}{12}=\frac{1}{12} \) (the dropdown shown has \( -\frac{1}{12} \), which is wrong).

Multiplicative Inverses:

1. The Multiplicative Inverse of \( 4=\frac{1}{4} \) (correct as per the dropdown shown).
2. The Multiplicative Inverse of \( -9=-\frac{1}{9} \) (correct as per the dropdown shown).
3. The Multiplicative Inverse of \( \frac{2}{3}=\frac{3}{2} \) (the dropdown shown has \( \frac{2}{3} \), which is wrong).
4. The Multiplicative Inverse of \( -\frac{1}{12}=-12 \) (the dropdown shown has \( \frac{1}{12} \), which is wrong).

Answer:

Additive Inverses:

1. Correct: \( -4 \)
2. Correct: \( 9 \)
3. Incorrect, should be \( -\frac{2}{3} \)
4. Incorrect, should be \( \frac{1}{12} \)

Multiplicative Inverses:

1. Correct: \( \frac{1}{4} \)
2. Correct: \( -\frac{1}{9} \)
3. Incorrect, should be \( \frac{3}{2} \)
4. Incorrect, should be \( - 12 \)

(If the problem is to select the correct dropdowns, the correct selections are:

Additive Inverse of \( \frac{2}{3} \): \( -\frac{2}{3} \)

Additive Inverse of \( -\frac{1}{12} \): \( \frac{1}{12} \)

Multiplicative Inverse of \( \frac{2}{3} \): \( \frac{3}{2} \)

Multiplicative Inverse of \( -\frac{1}{12} \): \( - 12 \))