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To effectively convey the concept of inequality graphing, imagine it as a visual representation of a range of numbers that satisfy the given inequality. When we graph inequalities like the one from our exercise, we are looking for all the possible values of x that will make the inequality true.

Using a number line is a common way to illustrate these values. For the inequality x ≤ 7, we mark the point corresponding to 7 with a closed dot, which tells us that 7 is a part of the solution. The line that extends from this point to the left signifies all numbers that are less than 7, which also satisfy the inequality. It's this combination of the dot and line that provides a clear visual cue to anyone interpreting the graph.

The number line representation is a fundamental tool for illustrating mathematical concepts and solutions. For inequalities, it serves as a straightforward method to visualize the solution set. The values on a number line are arranged with lower numbers to the left and higher numbers to the right.

When representing inequalities like x ≤ 7, placing a closed circle on the 7 indicates that this number is included in the solution. If the inequality did not include the number 7 itself—represented as x < 7—we would use an open circle instead. Then, a horizontal line or arrow is drawn from the circle to cover all possible values satisfying the condition, providing a simple yet effective visual summary.

The process of algebraic manipulation involves rearranging and simplifying expressions or inequalities without altering their meaning or solution sets. In our example, we start with the original inequality 3x + 4 ≤ 2x + 7, and our goal is to isolate the variable x on one side. We achieve this by performing operations such as subtracting 2x from both sides. This step is based on the fundamental principle that what we do to one side of an inequality, we must do the same to the other side to maintain balance.

Once simplified, we end up with x ≤ 7. This is a form of the inequality that is straightforward to understand and graph. It's important to know these techniques for algebraic manipulation as they are used extensively for solving a variety of mathematical problems.