91Ó°ÊÓ

Algebra is a branch of mathematics dealing with symbols, like letters, that represent numbers. In inequalities, these symbols commonly serve as variables, representing values that we need to find or solve for. The inequalities can be seen as the balance between two sets of expressions involving these variables.

An important point in algebra is manipulating these expressions to find the range of values that satisfy the inequality. In this exercise, the inequality given is: \( 5(x+4) > 5x + 10 \) The goal here is to discover the values of \( x \) that make this inequality true. For the task at hand, simplifying both sides using algebraic techniques allows us to see what values the variable \( x \) can hold.

Real numbers include all the numbers on the number line: integers, fractions, and even irrationals like \( \sqrt{2} \) and \( \pi \). When solving inequalities, any solution often refers to an interval or set of real numbers that satisfy the condition.

In this specific exercise, after we've simplified the inequality and eliminated the variable \( x \), we get a true statement: \( 20 > 10 \). This does not specify a particular number but indicates that the inequality holds for all real numbers. As a result, \( x \) can indeed be any real number, showing the importance of understanding the broader context of real numbers upon solving inequalities.

The distributive property is a useful, fundamental property in algebra that helps us to simplify expressions and solve equations or inequalities. It states:\[ a(b + c) = ab + ac \] Applying this property allows us to multiply a single term across a sum or difference within parentheses.

For this exercise, the property is applied to the left side of the inequality: \( 5(x + 4) \) This becomes \( 5 \times x + 5 \times 4 \), or \( 5x + 20 \). By doing so, the inequality is transformed into a more manageable expression, highlighting the two sides clearly to proceed with further operations, such as subtraction or factoring, to solve the inequality.