Question

Fill in the blanks. (i) If $x > y$ and $z < 0$, then $-xz$ $-yz$. (ii) If $p > 0$ and $q < 0$, then $p - q$ $p$. (iii) If $-2x + 1 \ge 9$, then $x$ $-4$.

• A. a
• B. b
• C. c
• D. d

Answer 1

The Correct Option is D

Let's analyze and solve each part of the problem step-by-step:

1. Consider the statement: If \(x > y\) and \(z < 0\), then \(-xz \, \_\_\_ \, -yz\). 

Explanation:

• If \(z < 0\), multiplying an inequality by a negative number reverses the inequality sign.
• Start with the inequality \(x > y\).
• When you multiply both sides by \(-z\) (which is positive since \(z\) is negative):
  • \(-xz < -yz\) because multiplying by a negative reverses the inequality.
• Therefore, the correct word for the blank is "less than" (denoted by \(<\)).

1. Consider the statement: If \(p > 0\) and \(q < 0\), then \(p - q \, \_\_\_ \, p\).

Explanation:

• Since \(q < 0\), subtracting \(q\) is equivalent to adding a positive number.
• Therefore, \(p - q = p + (-q) > p\) because you are adding a positive value (the absolute value of \(q\)).
• Hence, the correct word for the blank is "greater than" (denoted by \(>\)).

1. Consider the statement: If \(-2x + 1 \ge 9\), then \(x \, \_\_\_ \, -4\).

Explanation:

• Solve the inequality for \(x\):
  • Start with \(-2x + 1 \ge 9\).
  • Subtract 1 from both sides: \(-2x \ge 8\).
  • Divide both sides by \(-2\), and reverse the inequality sign: \(x \le -4\).
• Thus, the correct word for the blank is "less than or equal to" (denoted by \(-4\)).

In summary, the correct answers for the blanks are:

1. \(-xz < -yz\) (less than)
2. \(p - q > p\) (greater than)
3. \(x \le -4\) (less than or equal to)

Therefore, the complete set of blanks with the correct operations filled is:

• \(-xz < -yz\)
• \(p - q > p\)
• \(x \le -4\)