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Chapter 2: Problem 74

Solve using the addition and multiplication principles. $$ 8-2 y \geq 14 $$

Short Answer

Expert verified
y \leq -3

Step by step solution

01

- Isolate the variable term

First, subtract 8 from both sides of the inequality.\[ 8 - 2y - 8 \geq 14 - 8 \] which simplifies to: \[ -2y \geq 6 \]
02

- Solve for y

Next, divide both sides of the inequality by -2. Keep in mind, dividing by a negative number reverses the inequality symbol. \[ \frac{-2y}{-2} \leq \frac{6}{-2} \] which simplifies to: \[ y \leq -3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

addition principle
One of the crucial tools in solving inequalities is the addition principle. This principle allows you to add or subtract the same quantity on both sides of the inequality without changing its direction. This means if you have an inequality like 8 - 2y 鈮� 14, you can subtract 8 from both sides to help isolate the variable term.

Let's see how this applies to our exercise:
  • First, we subtract 8 from both sides: \[ 8 - 2y - 8 \ge 14 - 8 \] which simplifies to: \[ -2y \ge 6 \]
Notice how the equality symbol remains the same while simplifying the inequality involves straightforward arithmetic operations. This step helps isolate the variable term on one side, making further calculations much easier.
multiplication principle
Next, we use the multiplication principle, which states that you can multiply or divide both sides of the inequality by the same non-zero number. However, it's essential to remember a specific rule when dealing with inequalities and multiplication:

When you multiply or divide both sides by a negative number, you must reverse the inequality symbol.
  • Continuing our problem: \[ -2y \ge 6 \] we need to divide both sides by -2 to isolate 'y'. This division will reverse the inequality sign: \[ \frac{-2y}{-2} \le \frac{6}{-2} \] which simplifies to: \[ y \le -3 \]
Upon completing the division by -2, the inequality symbol changes from 鈮� to 鈮�, showing how critical it is to remember this reversal rule.
reversing inequality symbol
Reversing the inequality symbol occurs when you multiply or divide both sides of an inequality by a negative number. This rule is essential and often overlooked. Here鈥檚 why it matters:
  • In our exercise: \[ -2y \ge 6 \], dividing by -2 not only simplifies the variable term but also affects the inequality symbol. Instead of: \[ \frac{-2y}{-2} \ge \frac{6}{-2} \], it becomes: \[ y \le -3 \]
So, why does the reversal happen? Imagine flipping the number line: multiplying or dividing by a negative shifts the direction. This change ensures the inequality correctly reflects the new relationships between numbers. Remembering and applying this rule correctly is key to solving inequalities accurately.

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