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

Question

43d≥−20

34(43)d≥−20(34)

d≥−15

(1 point)
Responses

Multiplication Property of Inequality
Multiplication Property of Inequality

Subtraction Property of Inequality
Subtraction Property of Inequality

Division Property of Inequality
Division Property of Inequality

Addition Property of Inequality

Answer 1

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

This is because the inequality \(43d \geq -20\) is being multiplied by 34, which maintains the direction of the inequality since 34 is a positive number. Therefore, the applicable property here is the Multiplication Property of Inequality.

Answer 2

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

Step 1:

−6(3w−7)>−66

−18w+42>−66

Step 2:

−18w+42−42>−66−42

−18w>−108

Step 3:

−18w−18>−108−18

w<6

(1 point)
Responses

Multiplication Property of Inequality
Multiplication Property of Inequality

Division Property of Inequality
Division Property of Inequality

Subtraction Property of Inequality
Subtraction Property of Inequality

Addition Property of Inequality

Answer 3

The property of inequality that justifies the math in Step 2 is the Subtraction Property of Inequality.

In Step 2, you subtract 42 from both sides of the inequality \(-18w + 42 > -66\), which maintains the direction of the inequality. Therefore, the applicable property here is the Subtraction Property of Inequality.

Answer 4

Juana, Anish, and Giselle are comparing their solutions to the inequality 56x≤−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

Juana and Anish
Juana and Anish

Giselle and Anish
Giselle and Anish

Giselle and Juana
Giselle and Juana

Anish, Juana, and Giselle

Answer 5

To solve the inequality \(56x \leq -10\), we will divide both sides by 56 to isolate \(x\). This leads to:

\[ x \leq \frac{-10}{56} \]

Since we are dividing by a positive number (56), the direction of the inequality does not change. This means that Giselle is correct in stating that the inequality symbol must stay the same.

Juana is incorrect because there is no need to reverse the inequality symbol just because there is a negative sign on the right side.

Anish is correct because the Division Property of Inequality is applied when dividing both sides of the inequality by a positive number.

Therefore, the correct combination is Giselle and Anish. They are both correct in their reasoning regarding the inequality.

Answer 6

Use the image to answer the question.

A number line ranges from negative 12 to 0 in increments of 1. A closed point is plotted at negative 7. An arrow extends indefinitely left from the closed point past negative 12 along the number line.

What inequality does the number line graph?

(1 point)
Responses

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

x<−7
x less than negative 7

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

x>−7

Answer 7

The number line shows a closed point at \(-7\) and an arrow extending indefinitely to the left, which indicates that all numbers less than or equal to \(-7\) are included in the solution.

Therefore, the inequality that the number line graph represents is:

\(x \leq -7\) (x less than or equal to negative 7).