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When solving inequalities, one of the most pivotal steps is isolating the variable. This means getting the variable on one side of the inequality sign by itself. For our example, the inequality is \(6x + 2 < 18\).

To begin, we need to remove any numbers that are not connected to the variable. Here, we subtract 2 from both sides of the inequality. This keeps the equation balanced, similar to an old-fashioned scale.

If we subtract 2 on the left side, we should also do it on the right side, resulting in \(6x + 2 - 2 < 18 - 2\). This simplifies to \(6x < 16\).

By isolating the term \(6x\), we've completed the first step towards finding the solution. This method is crucial in solving not just inequalities, but equations in general.

After isolating the variable, the next step is to solve the inequality. In our example, we have reduced it to \(6x < 16\). The key focus now is to get 'x' by itself on one side of the inequality.

We do this by dividing both sides by the coefficient of \(x\), which in this case is 6. Dividing both sides keeps the inequality balanced, just like subtracting did. Thus, we have:

• \(\frac{6x}{6} < \frac{16}{6}\)

This simplifies further to \(x < \frac{8}{3}\).

It’s important to remember that when you divide (or multiply) both sides of an inequality by a negative number, you need to flip the inequality sign. However, in our example, we are dividing by a positive number, so the inequality sign remains the same.

In the case of \(6x < 16\), the inequality solution \(x < \frac{8}{3}\) means that the variable \(x\) can be any number less than \(\frac{8}{3}\). This describes a range of possible solutions.

The process of solving inequalities is very similar to solving equations, and following a step-by-step approach is very helpful. It serves as a plan or a guide that leads you towards finding the correct answer. In our example, the steps were straightforward yet effective.

We began by isolating the variable, which involved removing any added or subtracted numbers from the variable side. Then we divided both sides by the coefficient of the variable, ensuring the inequality was balanced throughout.

Here’s a summary of what we did:

• Subtract 2 from both sides to simplify \(6x + 2 < 18\) to \(6x < 16\).
• Divide both sides by 6, to isolate \(x\), resulting in \(x < \frac{8}{3}\).

These steps showcase the simplicity and power of step-by-step algebra. By breaking down the problem into smaller, manageable parts, solutions to inequalities become clear and systematic. This logical approach can be applied not just to inequalities, but to many algebraic problems as well, making it a valuable skill in mathematics.