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Solving inequalities is a core part of elementary algebra. It involves finding all values of a variable that make an inequality true. Unlike equations where you find specific solutions, inequalities describe a range of possible solutions.

When solving inequalities, the steps are very similar to solving equations. However, there are some crucial differences, like the direction of the inequality sign.

For example, when multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This rule is important and different from solving equations. Keeping these differences in mind helps you solve inequalities correctly.

• Subtraction or addition of terms to both sides maintains the inequality's direction.
• Multiplication or division by a positive number also maintains the inequality's direction.
• Multiplication or division by a negative number reverses the inequality's direction.

To solve for a variable in an inequality, you typically need to isolate the variable on one side. This process is known as the isolation of variables. Here’s how you can do it:

Let's start with the inequality: \(2y > -7\).

Your goal is to get the variable y alone on one side. In this case, y is currently multiplied by 2. The easiest way to isolate y is by performing the inverse operation of multiplication, which is division.

You divide both sides of the inequality by the coefficient of y, which is 2. This operation simplifies the inequality, allowing you to solve for the variable easily. This step-by-step isolation of the variable is essential to solving inequalities correctly.

Understanding the division of inequalities is crucial when solving problems like the one we are looking at:
Starting with: \(2y > -7\),

You divide both sides by the coefficient of y, which is 2: \br> \(\frac{2y}{2} > \frac{-7}{2}\).

When dividing by a positive number, the inequality sign remains the same. However, if we divided by a negative number, we would need to reverse the inequality sign. This is a critical rule in inequalities to remember.

After simplifying, you get: \(y > - \frac{7}{2}\). By doing this, you have isolated the variable y, completing the division part of solving the inequality.
Every time you simplify an inequality, double-check for positive and negative operations to ensure the inequality sign is correct.

• When dividing by a positive number, the direction of the inequality stays the same.
• When dividing by a negative number, always reverse the inequality sign.