91Ó°ÊÓ

In mathematics, especially when dealing with polynomial expressions or inequalities like the one in our exercise, combining like terms is a fundamental step to simplify the expression. Like terms are terms that have the same variables raised to the same power, but they might have different coefficients. In the expression \(6x + 3 - 8x > -3\), the like terms are \(6x\) and \(-8x\). These terms both contain the variable \(x\). By combining these, we add their coefficients:

• \(6x - 8x = -2x\)

This simplifies the inequality to \(-2x + 3 > -3\). This step reduces the expression to fewer terms, making it easier to handle in subsequent steps. Simplification is crucial as it helps in getting clearer insights into what the inequality represents.

When solving inequalities, isolating the variable term is an essential step. It often involves moving terms from one side of the inequality to the other, allowing us to focus on just the variable we want to solve for. In our exercise, after we simplify the expression to \(-2x + 3 > -3\), the next goal is to isolate the \(-2x\) term.To do this, we subtract 3 from both sides of the inequality:

• Subtract 3 from the left: \(-2x + 3 - 3\)
• Subtract 3 from the right: \(-3 - 3\)

After simplifying, we get \(-2x > -6\).The idea is to have the variable term alone on one side of the inequality. This is similar to isolating a variable in an equation but requires extra care because we're dealing with inequalities which have different rules, especially when it comes to multiplication or division by negative numbers.

Solving inequalities involves getting the variable by itself on one side of the inequality sign. In our problem, we have reached \(-2x > -6\). To solve for \(x\), we need to divide both sides by \(-2\). A key point in inequalities is that whenever you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed.So dividing both sides by \(-2\), we have:

• \(-2x / -2 < -6 / -2\)

Simplifying this gives \(x < 3\).It's crucial to remember the rule about reversing the inequality sign because failing to do so can lead to incorrect solutions. This rule distinguishes inequality operations from equations and is a vital concept when solving inequalities.