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

The given steps are: $\frac{4}{3}d \geq -20$ Multiplying both sides by $\frac{3}{4}$ gives: $\frac{3}{4} \times \frac{4}{3} d \geq -20 \times \frac{3}{4}$ Simplifying: $d \geq -15$ This step involves multiplying both sides of the inequality by $\frac{3}{4}$, so the correct justification is A. Multiplication Property of Inequality.

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

Let's analyze the steps:

• Step 1: $-6(3w - 7) > -66$ $-18w + 42 > -66$
• Step 2: Subtracting 42 from both sides: $-18w + 42 - 42 > -66 - 42$ Simplifying: $-18w > -108$

The operation performed in Step 2 is subtracting 42 from both sides. This is justified by D. Subtraction Property of Inequality.

3. Juana, Anish, and Giselle's statements about the inequality $\frac{5}{6}x \leq -10$.

• Juana says the inequality symbol must be reversed because of the negative sign in the inequality. This is incorrect because merely having a negative sign does not automatically reverse the inequality unless we multiply or divide by a negative number.
• Anish says the Multiplication Property of Inequality or Division Property of Inequality must be applied. This is correct, as we will need to divide both sides by $\frac{5}{6}$ or multiply by $\frac{6}{5}$.
• Giselle says the inequality symbol must stay the same. This is also correct, since in this case, we are not multiplying or dividing by a negative number.

Thus, D. Giselle and Anish are correct.

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Would you like any further details or clarifications on these concepts? Here are 5 related questions to explore:

1. How does multiplying or dividing by a negative number affect inequalities?
2. What are the steps to solve $\frac{5}{6}x \leq -10$?
3. How do the properties of inequality apply to compound inequalities?
4. What is the difference between the addition and subtraction properties of inequality?
5. How do we check solutions to inequality problems?

Tip: Always pay attention to the sign of the coefficient when solving inequalities, especially when multiplying or dividing by negative values!