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When solving inequalities or equations, one of the first steps is often to simplify expressions by combining like terms. This means aggregating terms that have the same variables raised to the same power. In our given inequality \( \frac{x}{5} - \frac{x}{2} \leq 1 \), we see two terms with the variable \( x \): \( \frac{x}{5} \) and \( -\frac{x}{2} \).

To combine these terms, we need a common denominator, which helps in making the fractions easily comparable and combinable. The denominators here are 5 and 2. A common denominator for 5 and 2 is 10. We convert:

• \( \frac{x}{5} \to \frac{2x}{10} \) by multiplying both the numerator and denominator by 2
  
• \( \frac{x}{2} \to \frac{5x}{10} \) by multiplying both the numerator and denominator by 5

Once the fractions are expressed with the same denominator, they can be combined: \( \frac{2x}{10} - \frac{5x}{10} = \frac{-3x}{10} \). This simplification is crucial for solving the inequality as it reduces complexity and clarifies the variable's role.

Graphing inequalities helps us visually represent the set of solutions on a number line, making it easier to interpret. In the inequality solution \( x \geq -\frac{10}{3} \), we require understanding its graphical representation.

When graphing, one must remember that:

• A closed dot or circle represents that the number at that point is included in the solution, due to the "equal" part in \( \geq \).
  
• An open dot would be used for "less than" (<) or "greater than" (>) without equality involved.
• The shaded line indicates all possible values that satisfy the inequality.

Here, the closed dot is placed at \( -\frac{10}{3} \), showing this value is part of the solution. From \( -\frac{10}{3} \), the line is shaded to the right, indicating that all values greater than or equal to \( -\frac{10}{3} \) are solutions. This visual tool clarifies the extent and boundary of solutions.

The process of isolating a variable in an inequality means getting the variable alone on one side. For instance, in transforming \( \frac{-3x}{10} \leq 1 \) into \( x \geq -\frac{10}{3} \), isolating \( x \) is essential.

To isolate \( x \), first multiply each term of the inequality by the reciprocal of \( \frac{-3}{10} \), which is \( \frac{-10}{3} \). This step undoes the division within \( \frac{-3x}{10} \), effectively moving \( x \) out of the fraction:
\[ \frac{-3x}{10} \times \frac{-10}{3} \, \leq \, 1 \times \frac{-10}{3} \] results in: \( x \geq -\frac{10}{3} \). It's crucial to remember that multiplying or dividing an inequality by a negative number flips the inequality sign. This pivotal rule ensures that we accurately represent the relationship between expressions once the variable is isolated. Thus, understanding this principle ensures no misinterpretation occurs during the solution process.