Question

use the correct inequality symbol to complete each statement. a. if c < 0 and a < b, then ac \boxed{<} bc. b. if c < 0 and a < b, then a - c \boxed{\blacktriangledown} b - c. c. if c > 0 and a < b, then \frac{a}{c} \square

Explanation:

Part A

Step1: Recall inequality property (multiplying by negative)

When multiplying both sides of an inequality \(a < b\) by a negative number \(c<0\), the inequality sign reverses. So \(a\times c> b\times c\) (since \(c\) is negative). Wait, but the original answer in the box was \(<\), that was a mistake. Wait, no, let's recheck. Wait, the problem says "If \(c < 0\) and \(a < b\), then \(ac\) [ ] \(bc\)". The correct property: when you multiply both sides of an inequality by a negative number, the inequality sign flips. So \(a < b\), multiply both sides by \(c<0\), we get \(ac>bc\). But maybe the original box was wrong, but let's proceed with the correct steps. Wait, maybe I misread. Wait, the user's image shows a box with \(<\) for part A, but that's incorrect. Let's do it properly.

Step1: Identify the inequality rule for multiplication by negative

Given \(a < b\) and \(c<0\) (negative). The rule is: if \(a < b\) and \(c<0\), then \(ac>bc\) (because multiplying by a negative reverses the inequality). But maybe the user's initial box was incorrect, but let's solve each part.

Part B

Step1: Recall inequality property (subtracting a number)

The property of inequalities: if \(a < b\), then \(a - c < b - c\) for any real number \(c\). Because subtracting the same number from both sides doesn't change the inequality direction. Let's verify: \(a < b\), subtract \(c\) from both sides: \(a - c < b - c\). So the symbol is \(<\).

Part C

Step1: Recall inequality property (dividing by positive)

When dividing both sides of an inequality \(a < b\) by a positive number \(c>0\), the inequality sign remains the same. So \(\frac{a}{c}<\frac{b}{c}\) (since \(c\) is positive, dividing doesn't reverse the inequality).

Final Answers:

A. The correct symbol should be \(>\) (but if we follow the problem's initial box which was wrong, but the correct step: \(a < b\), \(c<0\), so \(ac>bc\))
B. \(a - c < b - c\), so symbol is \(<\)
C. \(\frac{a}{c}<\frac{b}{c}\), so symbol is \(<\)

But let's present each part with step - by - step:

Part A

Answer:

\(>\)

Part B