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

Question

4/3d≥−20

3/4(4/3)d≥−20(3/4)

d≥−15

Responses:

Subtraction Property of Inequality

Multiplication Property of Inequality

Addition Property of Inequality

Division 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 this case, the equation \( \frac{3}{4} \left( \frac{4}{3} d \right) \geq -20 \left( \frac{3}{4} \right) \) involves multiplying both sides of the inequality by \( \frac{3}{4} \). Since \( \frac{3}{4} \) is a positive number, the direction of the inequality does not change. Therefore, 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

−18w+42>−66

Step 2:

−18w+42−42>−66−42

−18w>−108

Step 3:

−18w/−18>−108/−18

w<6
Responses:

Addition Property of Inequality

Division Property of Inequality

Subtraction Property of Inequality

Multiplication 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 are subtracting 42 from both sides of the inequality:

\[ −18w + 42 - 42 > -66 - 42 \]

This manipulation is justified by the Subtraction Property of Inequality, which states that if you subtract the same number from both sides of an inequality, the inequality remains true.

Therefore, the correct response is:

Subtraction Property of Inequality.

Answer 4

Juana, Anish, and Giselle are comparing their solutions to the inequality 5/6x≤−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?

Responses:

Juana and Anish

Anish, Juana, and Giselle

Giselle and Anish

Giselle and Juana

Answer 5

To solve the inequality \( \frac{5}{6}x \leq -10 \), we need to multiply or divide by a number. Since we are multiplying or dividing by a positive number (in this case, \( \frac{5}{6} \)), the direction of the inequality will remain the same.

• Juana is incorrect because the presence of a negative sign in the inequality does not require reversing the inequality symbol; that would only apply if we were multiplying or dividing by a negative number.
• Anish is correct because once we isolate \( x \), we can apply either the Multiplication Property or the Division Property of Inequality.
• Giselle is also correct because the inequality symbol indeed stays the same when we divide by a positive number.

Based on this analysis, the correct combination of people is:

Giselle and Anish.