91Ó°ÊÓ

When encountering problems with inequalities, the primary goal is to solve for a variable, in this case, the variable is \( x \). An inequality solution refers to the set of all possible values of \( x \) that make the inequality true. In our example, the compound inequality \( -6 < x-4 \leq 1 \) requires us to find a set of real numbers for \( x \) that satisfy both conditions simultaneously.

The process usually involves isolating the variable, as demonstrated in both parts of the compound inequality. After adding or subtracting numbers from both sides of each inequality, we end up with two simple inequalities: \( -2 < x \) and \( x \leq 5 \). Now we must look for the overlap, which represents the numbers that satisfy both conditions. These overlapping values are the 'solutions' to the compound inequality. Visualizing these solutions on a number line can make understanding the concept even clearer.

An algebraic expression is a collection of numbers, variables (like \( x \)), and operations (such as addition and multiplication) that represent a specific mathematical relationship. In our exercise, the expression \( x-4 \) inside the compound inequality is a simple algebraic expression.

To solve an inequality involving an algebraic expression, one typically performs a series of operations to isolate the variable. In this case, we added 4 to both sides of each part of the compound inequality to separate \( x \) from the constant number -4. This manipulation requires an understanding of the properties of equality and inequality, which govern how you can legally 'move' parts of an expression from one side of an equation or inequality to the other without changing the true nature of the relationship.

The discussion of real numbers is essential when solving algebraic inequalities because the solutions almost always lie in this set. Real numbers include all the rational and irrational numbers, meaning they can be whole numbers, fractions, decimals, and even non-repeating, non-terminating decimals like \(\pi\).

In our compound inequality \( -6 < x-4 \leq 1 \), the solution consists of all the real numbers between -2 and 5, including 5 (expressed as \( -2 < x \leq 5 \)). The solution does not include -2; this is important when graphing the solution or representing it in interval notation. In more complex inequalities, there may be exclusions or multiple intervals, but the solutions will always rest within the set of real numbers.