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Algebra is all about finding the values of unknown variables. When we solve inequalities like \(-6 < 3(x+2) < 9\), it means we're looking for a range of values for \(x\) that make the inequality true. This problem involves manipulating expressions using algebraic methods, such as distribution and solving for a single variable.

The key steps when dealing with an inequality in algebra include:

• Distribution: Apply multiplication across terms in parentheses.
• Combining like terms: Simplify the expression by collecting similar terms on one side.
• Balancing the inequality: Whatever operation you do on one part, you need to do across the entire inequality.

Keeping everything balanced helps maintain the correctness of the inequality throughout your solution process. As in algebra, the moves we make to isolate \(x\) must be mirrored in each section of the inequality.

A number line is a simple way to visualize the solutions to an inequality. For our solution \(-4 < x < 1\), you'll place values on a horizontal line to show where \(x\) can lie.

Here’s how to use a number line:

• Identify bounds: Identify the points \(-4\) and \(1\) on the line.
• Open circles: Use open circles at \(-4\) and \(1\) because these are not included in the solution. \(x\) is greater than \(-4\) and less than \(1\), but not equal to them.
• Shading: Shade the region between \(-4\) and \(1\) to indicate that \(x\) can be any value in this range.

This graphical approach makes it easier to understand and communicate the range of solutions, as it gives a clear visual cue on where \(x\) can be found.

Solving inequalities is similar to solving equations, but with a few twists. The goal is to isolate the variable on one side while respecting the inequality signs. Here’s how we solved \(-6 < 3(x+2) < 9\):

First, we distributed the \(3\) across \(x+2\), leading to \(-6 < 3x + 6 < 9\). This step uses the distributive property to eliminate parentheses.

Next, subtraction was used to remove \(6\) from each side, simplifying the inequality to \(-12 < 3x < 3\). This operation helps zero in on the \(x\).

Finally, dividing each part of the inequality by \(3\) simplified it to \(-4 < x < 1\). At every step, we made sure to maintain the inequality by applying operations uniformly. If we had multiplied or divided by a negative number, we’d flip the inequality signs, but that wasn’t necessary here.

Remember: Inequalities show a range of possible solutions, and each step in isolation brings clarity to what values \(x\) can take.