91Ó°ÊÓ

When solving inequalities, an essential tool is the addition principle. This principle states that you can add or subtract the same value from both sides of an inequality without changing its truth.
In our exercise, the first step involves subtracting 5 from both sides of the inequality. This keeps the inequality balanced. Here's why it works:

• Original Inequality: \[ 5 - 7y \geq 5 \]
• Subtract 5: \[ 5 - 7y - 5 \geq 5 - 5 \]
• Result: \[ -7y \geq 0 \]

This step simplifies the inequality, making it easier to solve for y. By using the addition principle, we just changed our inequality to a simpler form without altering its meaning.

Next, let's explore the multiplication principle, another fundamental concept in solving inequalities. The multiplication principle states that you can multiply or divide both sides of an inequality by the same non-zero number. However, there's an important caveat: if you multiply or divide by a negative number, you must reverse the inequality sign.
In our problem, after simplifying with the addition principle, we had: \[ -7y \geq 0 \]. To solve for y, we need to isolate it by dividing both sides by -7. Because we are dividing by a negative number, we must reverse the inequality sign:

• Original: \[ -7y \geq 0 \]
• Divide both sides by -7: \[ y \leq 0 \]

This step gives us the final solution for the inequality. Using the multiplication principle correctly is crucial for ensuring the solution is accurate.

One crucial aspect of solving inequalities is knowing when to reverse the inequality sign. This usually occurs during the multiplication principle when you multiply or divide both sides of an inequality by a negative number. This step is vital since ignoring it can lead to incorrect solutions.
For instance, in the provided exercise, we reach the point \[ -7y \geq 0 \]. To isolate y, we divide both sides by -7. Here's a breakdown:

• Inequality: \[ -7y \geq 0 \]
• Divide by -7: \[ y \leq 0 \]

Notice we reversed \< \geq> to \< \leq>. This reversal is essential because dividing by a negative flips the direction of the inequality.
Never forget to reverse the sign when working with negative numbers in inequalities, ensuring your solution stays accurate and valid.