91Ó°ÊÓ

The addition principle is used when you want to isolate the variable by eliminating terms from one side of the inequality. For example, in the exercise given, we need to move all variable terms to one side to simplify the inequality.

Here are the steps:

• Identify the terms you need to move.
• Add the same value to both sides of the inequality. This keeps the inequality balanced.

In our example:
We started with: \(7-9y \leq 4-8y\)

To isolate terms with y on one side, add \(8y\) to both sides:
\[7 - 9y + 8y \leq 4 - 8y + 8y\]

When simplified, this becomes:
\[7 - y \leq 4\]

Through these steps, we use the addition principle to simplify the inequality efficiently.

The multiplication principle is a key tool when working with inequalities, especially when you need to solve for a variable. One thing to remember is if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Let's continue with our example:

After isolating the term containing y, we had:
\[ - y \leq -3\]

To solve for y, we need to get rid of the negative sign. We do this by multiplying both sides by \(-1\):

\( -1 \times (- y ) \geq -1 \times ( -3 ) \)

Notice how the inequality sign reverses when both sides are multiplied by a negative number. This is an important rule to remember as it ensures the inequality remains true. The inequality now simplifies to:
\[ y \geq 3\]

The multiplication principle helps us finally isolate the variable and find the solution to the inequality.

Isolating the variable in an inequality involves getting the variable term by itself on one side of the inequality. This process typically involves using both the addition and multiplication principles.

In the initial inequality:
\(7 - 9y \leq 4 - 8y\),
we first used the addition principle to move variable terms to one side:
\[7 - 9y + 8y \leq 4 - 8y + 8y\],
which leaves us with:
\[7 - y \leq 4\]

Next, we isolate y by performing operations to remove constants and coefficients. Since we wanted to remove the 7, we subtracted 7 from both sides:
\[7 - y - 7 \leq 4 - 7\],
simplifying to:
\[- y \leq -3\]

Finally, we applied the multiplication principle by multiplying both sides by -1 (and reversing the inequality sign) to isolate y:
\( -1 \times - y \geq -1 \times -3 \),
giving us the solution:
\[ y \geq 3 \]

Isolating the variable is a critical step in solving inequalities, requiring both strategic addition/subtraction and multiplication/division while adhering to the rules of inequalities.