91Ó°ÊÓ

When working with algebraic inequalities, we are dealing with mathematical expressions that show the relationship between two values, indicating that one value is greater than, less than, greater than or equal to, or less than or equal to another value. Unlike equations, inequalities do not indicate exact equality but rather a range of possible solutions. For example, the inequality from the exercise, \(3x - 18 > 0\), tells us that the value of \(3x - 18\) is greater than 0.

Algebraic inequalities can have one or more variables and can come in various forms, such as linear, quadratic, or rational inequalities. Solving them typically involves using similar techniques as one would use for equations, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number—while paying special attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers.

To isolate the variable in an inequality, similar principles apply as when solving standard algebraic equations. The main objective is to get the variable on one side of the inequality sign by itself. In the given exercise, the first step is to eliminate the '-18' on the left side, which is achieved by adding 18 to both sides, resulting in \(3x > 18\).

Importance of Reversing the Inequality

Keep in mind that reversing the inequality sign is necessary when multiplying or dividing by a negative number, which does not occur for this exercise but is crucial for solving certain inequalities. This step is fundamental because it reflects the property that if '\(a < b\)' and 'c' is negative, then '\(ac > bc\)'.

The solutions to an inequality are all the values that make the inequality true. For the inequality \(3x > 18\), once we've isolated the variable and determined that \(x > 6\), we have defined a set of possible values for 'x' that are a part of the solution set.

Unlike equations, which might have a single solution, inequalities generally have infinite solutions—every number greater than 6, in this case, is a valid solution. This range of answers is why it's imperative to understand that the solution to an inequality is not a single value but rather a set or an interval of numbers. To depict this, we often use number lines, graphs, or interval notation. Checking the solutions provided in the exercise, as shown in the original solution steps, confirms that only option (d) \(x = 11\) is correct because it falls within the solution set, where 'x' is greater than 6.