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The distributive property is a fundamental concept in algebra that plays a crucial role in simplifying expressions and solving equations. It allows us to multiply a single term by each term inside a set of parentheses. This property is expressed as:\[ a(b + c) = ab + ac \]In the context of solving inequalities, the distributive property helps in distributing multiplication over addition or subtraction within parentheses. In the exercise given, we used the distributive property to eliminate the parentheses and simplify the inequality:

• Given expression: \(2(z - 4) > 10z\)
• Distributed: \(2z - 8 > 10z\)

By applying this property, the expression inside the parentheses became more straightforward, paving the way for solving the inequality.

Isolating the variable is an essential step when solving equations or inequalities. The goal is to get the variable by itself on one side of the equation or inequality sign. This often involves a sequence of operations, such as addition, subtraction, multiplication, or division, applied to both sides of the equation.In our problem, we aimed to isolate the variable \(z\) by performing the following steps:

• We started with: \(2z - 8 > 10z\).
• Then, subtracted \(2z\) from both sides to remove it from the left: \(2z - 2z - 8 > 10z - 2z\).
• The inequality simplifies to: \(-8 > 8z\).

By isolating the variable, we pave the way towards a solution, allowing easier manipulation and calculation to find the variable's potential values.

Comparing algebraic expressions is a key component in solving inequalities. This involves examining and manipulating expressions to highlight relationships and find possible solutions. It requires careful attention to the inequality signs, ensuring that operations preserve the meaning of the inequality.For the problem solved, once the expressions \(-8 > 8z\) were reached through simplification and isolation, the next step was to compare:

• Divide each term by \(8\) to simplify the inequality.
• The comparison \(-1 > z\) or equivalently \(z < -1\) indicates the values that satisfy the inequality.

Effectively comparing expressions helps in determining the solution set of the variable, describing the range of values that meet the condition given by the inequality. This skill is notably beneficial for algebraic problem-solving as it allows a methodical approach to reaching final, meaningful results.