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Understanding how to represent inequalities on a graph is crucial for interpreting and communicating solutions effectively. Inequalities show a relationship where one value is not strictly equal to another, but rather less than, greater than, less than or equal to, or greater than or equal to.

For the inequality given in the exercise,
\(x \leq 1\), it specifies that the variable \(x\) can take any value that is less than or equal to 1. To visualize this on a graph, the number line is used. By plotting a specific point or line that corresponds to \(x = 1\), we create a visual reference point. Since the inequality includes 'less than or equal to', it is represented by a solid line or a filled-in dot on the number line to indicate that the value 1 itself is included in the set of solutions.

A number line is a straight, horizontal representation of numbers in a specific order, traditionally with smaller numbers to the left and larger numbers to the right. It serves as the basis for graphing inequalities because it provides a visual method for comparing magnitudes and understanding variable ranges.

Inequalities are mapped on number lines by highlighting or shading the range of numbers that satisfy the inequality. In the given exercise, \(x \leq 1\), the number line is used to plot the point where \(x = 1\). All points to the left of this indicate values of \(x\) that are less than 1, fleshing out the full set of solutions. The entire area from this point to the left is shaded, which is consistent with the meaning of 'less than or equal to'.

Vertical lines in a Cartesian coordinate system represent a single value for the x-coordinate across all y-values. They are particularly useful when graphing inequalities involving just one variable, like \(x\), as they visually display all the points where the x-value is constant.

When the inequality involves just 'x', such as \(x \leq 1\), plotting the vertical line at \(x = 1\) acts as the boundary of the solution set. This line is drawn passing through the x-axis at the point (1, 0). Since this inequality includes the value 1, the line is solid. If the inequality were 'less than' without including the equal part (\(x < 1\)), the depicted line would be dashed, signaling that points on the line are not included in the solution set.