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Inequalities are similar to equations, but instead of showing equality between two expressions, they show a relationship of greater than or less than. Like equations, inequalities can also be solved using algebraic manipulation. The key difference is the consideration of the inequality sign (<, >, ≤, ≥), which tells us the nature of the relationship between the two sides. Here, we are dealing with a linear inequality:

• The goal is to find the maximum or minimum value for the variable, for which the inequality holds true.
• When solving inequalities, it's important to remember that multiplying or dividing by a negative number reverses the inequality sign.

Starting with the inequality \(-6 + 5x < 19\), the process involves isolating the term with the variable on one side of the inequality through a series of step-by-step algebraic manipulations, eventually finding the range of values that satisfy the inequality. Understanding these steps assists in effectively solving any linear inequality.

Algebra problems involving inequalities require a systematic approach. These problems can be seen as puzzles that must be solved with logical steps:

• First, simplify both sides of the inequality as much as possible.
• Second, perform basic operations to isolate the variable term.
• Finally, solve for the variable while keeping track of any changes to the inequality sign.

With the problem isolating \(5x\) on one side required us to add 6 to both sides, giving us \(5x < 19 + 6\). Simplifying further to \(5x < 25\) set us up for the final steps. The systematic approach helps solve more complex algebraic problems by following similar steps. Remember to check your solution by plugging numbers into the original inequality to ensure they satisfy the condition.

Variable isolation is a fundamental step in solving both equations and inequalities. It involves getting the variable by itself on one side of the equation or inequality. The steps for isolating a variable typically involve:

• Reversing the order of operations: start with addition or subtraction, then proceed with multiplication or division.
• Ensuring every operation is applied to both sides of the equation or inequality to maintain balance.

In our inequality, simplifying \(5x < 25\) is achieved through division, and dividing both sides by 5 gives \(x < 5\). This results in isolating \(x\) to determine that any number less than 5 satisfies the inequality. Variable isolation guides us to the final answer, providing a clear understanding of what the inequality truly represents.