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Inequalities are mathematical expressions that describe a relationship of non-equality between two values. Solving an inequality means finding all possible values that make the inequality a true statement. Unlike equations, which have an exact solution, inequalities often have a range of solutions that satisfy the inequality condition.

In the exercise provided, the goal was to solve the inequality 6x - 10 ≤ -4. The step-by-step solution illustrates a systematic approach to finding this solution, which is x ≤ 1. This solution is interpreted as any number that is less than or equal to one makes the original inequality true. Understanding this concept is crucial in various fields, such as mathematics, economics, and engineering, where decision-making often involves evaluating options within certain constraints.

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to solve for a variable or to make comparisons. It involves a variety of techniques, such as combining like terms, factoring, expanding, and applying inverse operations. For instance, to solve the given inequality 6x - 10 ≤ -4, we utilized the property of adding the same number to both sides which keeps the inequality balanced.

Understanding the Balance

Similar to a scale in balance, whatever operation we perform on one side must be performed on the other to maintain the relationship. The original problem requires adding 10 to both sides, simplifying the inequality down to 6x ≤ 6.

Division in Inequalities

The next step involves dividing both sides by 6, which represents an inverse operation. This simplification yields a solution set that precisely describes all the values x can assume and still satisfy the original inequality.

Linear inequalities, such as the one in our exercise, represent regions on a number line rather than single points. These regions can be either open (not including the boundary point) or closed (including the boundary point) intervals.

An inequality that includes '≤' or '≥' symbols is indicative of a closed interval, meaning that the value represented by the inequality sign is a part of the solution set. In contrast, open intervals, denoted by '<' or '>', exclude the boundary value.

Graphical Representation

For a solution like x ≤ 1, graphically, we would draw a number line and shade the section to the left of the number 1, including the point at 1 which is often represented with a filled-in dot to show it's part of the solution set. An understanding of linear inequalities is essential in subjects that deal with ranges and differentials, such as calculus and statistics.