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A linear inequality is similar to a linear equation, but instead of an equal sign, it uses an inequality symbol. These symbols can include '<', '>', '\(\leq\)', or '\(\geq\)', indicating 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. The solution set of a linear inequality consists of all the values that satisfy the inequality.

For example, when solving the inequality \(18x + 45 \leq 12x - 8\), we are searching for all the possible values of \(x\) that make the inequality true. This set of values can be visualized on a number line and often represents a range of numbers. Visualizing the solution on a number line can be especially helpful for many students as it provides a clear picture of all the values that satisfy the inequality.

Algebraic manipulation involves rearranging and simplifying expressions using the principles of arithmetic and algebra. The goal is to isolate the variable we're solving for on one side of the inequality to find its possible values.

In the given example, subtracting \(12x\) and \(45\) from both sides represent algebraic manipulation steps to isolate the \(x\) variable. An important tip during manipulation is to perform the same operation on both sides of the inequality to maintain its balance. This step-by-step approach can be helpful, particularly for more complex inequalities, as it breaks down each phase of solving the inequality into more manageable parts.

To find the solution to an inequality, after algebraic manipulation, the final step typically involves dividing or multiplying both sides by a constant to solve for the variable. When doing so, it's crucial to remember that if we multiply or divide both sides of an inequality by a negative number, we need to reverse the inequality symbol.

In the exercise \( x \leq -53/6 \), we find that any number less than or equal to \(-\frac{53}{6}\) will satisfy the original inequality. Writing down the solution as a statement can aid in comprehension: 'The set of all numbers less than or equal to \(-\frac{53}{6}\) are solutions to the inequality.' This clarification can be invaluable for students who may not initially grasp why certain steps are necessary in finding the solution to an inequality.