What property of inequality justifies the steps shown in the following solution process?

Question

43d≥−20
4
3
𝑑
≥
−
20


34(43)d≥−20(34)
3
4
(
4
3
)
𝑑
≥
−
20
(
3
4
)


d≥−15
𝑑
≥
−
15


(1 point)
Responses

Division Property of Inequality
Division Property of Inequality

Addition Property of Inequality
Addition Property of Inequality

Multiplication Property of Inequality
Multiplication Property of Inequality

Subtraction Property of Inequality

Answer 1

The property of inequality that justifies the steps shown in the solution process is the Multiplication Property of Inequality.

In the solution, both sides of the inequality \( 43d \geq -20 \) are being multiplied by \( \frac{3}{4} \) to simplify the equation, which is an application of this property.

Thus, the correct response is:

Multiplication Property of Inequality.

Answer 2

What property of inequality can justify the math in Step 2?

Step 1:

−6(3w−7)>−66
−
6
(
3
𝑤
−
7
)
>
−
66


−18w+42>−66
−
18
𝑤
+
42
>
−
66


Step 2:

−18w+42−42>−66−42
−
18
𝑤
+
42
−
42
>
−
66
−
42


−18w>−108
−
18
𝑤
>
−
108


Step 3:

−18w−18>−108−18
−
18
𝑤
−
18
>
−
108
−
18


w<6
𝑤
<
6


(1 point)
Responses

Division Property of Inequality
Division Property of Inequality

Addition Property of Inequality
Addition Property of Inequality

Multiplication Property of Inequality
Multiplication Property of Inequality

Subtraction Property of Inequality

Answer 3

In Step 2, the operation performed is subtracting \( 42 \) from both sides of the inequality. This is an application of the Subtraction Property of Inequality, which states that if you subtract the same quantity from both sides of an inequality, the inequality remains true.

Thus, the correct response is:

Subtraction Property of Inequality.

Answer 4

Juana, Anish, and Giselle are comparing their solutions to the inequality 56x≤−10
5
6
𝑥
≤
−
10
. Juana says that the inequality symbol must be reversed because there is a negative sign in the inequality. Anish says the Multiplication Property of Inequality or Division Property of Inequality must be applied. Giselle says the inequality symbol must stay the same.

Based on these answers, which combination of people is correct?

(1 point)
Responses

Giselle and Anish
Giselle and Anish

Juana and Anish
Juana and Anish

Giselle and Juana
Giselle and Juana

Anish, Juana, and Giselle

Answer 5

To solve the inequality \( 56x \leq -10 \), we need to isolate \( x \) by dividing both sides by \( 56 \).

1. Juana's Statement: She incorrectly claims that the inequality symbol must be reversed just because there is a negative sign in the inequality. This is not correct because the inequality symbol is only reversed when multiplying or dividing both sides by a negative number, which is not the case here.

2. Anish's Statement: He correctly identifies that the Multiplication Property of Inequality or Division Property of Inequality must be applied to isolate \( x \). Since we are dividing both sides by a positive number (56), the inequality symbol will remain the same.

3. Giselle's Statement: Giselle claims that the inequality symbol must stay the same, which is also correct given the context of the operation being performed.

Therefore, the correct combination of people is:

Giselle and Anish.

Answer 6

Use the image to answer the question.

What inequality does the number line graph?

(1 point)
Responses

x≤−7
𝑥
≤
−
7
x less-than-or-equal-to negative 7

x>−7
𝑥
>
−
7
x greater than negative 7

x≥−7
𝑥
≥
−
7
x greater-than-or-equal-to negative 7

x<−7

Answer 7

Since I cannot see the image, I will provide guidance based on the descriptions commonly associated with number line graphs:

1. Closed Circle (●): If the number line graph has a closed circle at \(-7\), it indicates that \(-7\) is included in the solution. This corresponds to the inequalities:

   • \(x \leq -7\)
   • or \(x \geq -7\) depending on the direction of the shading (to the left or right).

2. Open Circle (○): If the number line graph has an open circle at \(-7\), it indicates that \(-7\) is not included in the solution. This corresponds to the inequalities:

   • \(x < -7\)
   • or \(x > -7\) depending on the direction of the shading.

Given these characteristics, please compare them with the graph to select the correct inequality that represents the number line you are analyzing. If you can describe the graph or its key features, I can assist you more accurately!

Answer 8

Graph x>11
𝑥
>
11
.(1 point)
Responses

Image with alt text: A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely left from the open point past 5 along the number line.

Image with alt text: A number line ranges from 5 to 15 in increments of 1. A closed point is plotted at 11. An arrow extends indefinitely left from the closed point past 5 along the number line.

Image with alt text: A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely right from the open point past 15 along the number line.

Image with alt text: A number line ranges from 5 to 15 in increments of 1. A closed point is plotted at 11. An arrow extends indefinitely right from the closed point past 15 along the number line.

Answer 9

To graph the inequality \( x > 11 \):

1. Open Point at 11: Since the inequality is "greater than" (not "greater than or equal to"), you should use an open circle at \( 11 \) to indicate that \( 11 \) itself is not included in the solution.

2. Arrow Extends Right: Since you want all numbers greater than \( 11 \), the arrow should extend indefinitely to the right from the open point.

Based on this description, the correct option that represents the graph of \( x > 11 \) is:

Image with alt text: A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely right from the open point past 15 along the number line.