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When we talk about inequality graphing, we're looking at a visual representation of all the possible solutions to an inequality. In our example, we've solved the inequality \( -2(x+3) \geq x \) to find that \( x \leq -2 \). To graph this on a number line, we start by drawing a horizontal line, which will serve as our number line. We then place a point on the number line where \( x = -2 \) because this is our boundary or 'cutoff' point.

Since the inequality includes the equal sign (\(\geq\)), we indicate this by drawing a closed dot on \( -2 \) to show that -2 is included in the set of solutions. From this point, we shade or draw an arrow to the left, because all other solutions are numbers less than -2. This visualization helps us see at a glance all the numbers that satisfy the inequality. The graph is an essential tool for understanding inequalities as it provides an intuitive way to interpret the solutions.

The heart of solving any algebraic inequality lies in algebraic manipulation. This involves moving terms and factors around an equation or inequality according to algebraic rules to isolate the variable in question. In our example, we start by distributing the -2 across \( x+3 \) to get \( -2x - 6 \geq x \). Next, we bring all terms containing \( x \) on one side by adding \( 2x \) to each side, resulting in \( -6 \geq 3x \).

Finally, dividing both sides by 3 gives us the isolated variable \( x \leq -2 \). Each of these steps—distribution, combining like terms, and dividing—serves the purpose of simplifying the inequality into a form where the variable stands alone and is easy to interpret. Mastering algebraic manipulation is crucial as it is a building block for solving more complex algebraic problems.

The number line representation is a powerful yet simple tool for illustrating numerical concepts, including inequalities. For the given inequality \( x \leq -2 \), the number line helps us represent the set of all real numbers that make the inequality true. It's done by first plotting a point at the critical value, which is -2 in this case.

Then, we decide on the appropriate symbol: a closed dot since \( x \leq -2 \) means \( -2 \) is part of the solution set. To represent that all numbers to the left of -2 are also solutions, we shade or draw a line or arrow extending left from the -2, showing that the solutions continue indefinitely in that direction. By doing this, anyone can quickly see which numbers satisfy the inequality. This method provides an immediate visual context that can be especially helpful for visual learners or when assessing a range of solutions at once.