91Ó°ÊÓ

Distributing in algebra is when you multiply a single term outside a set of parentheses across each term within the parentheses. This process adheres to the distributive property, which states that for any numbers a, b, and c, the expression \(a(b+c)\) equals \(ab + ac\).
In the original exercise, the term 4 is distributed across the expression inside the parentheses: \(4(x-2)\). This results in:
\[4(x) - 4(2) = 4x - 8\]\(4x - 8\) replaces the original set of parentheses. Distributing helps simplify the expression, making it easier to solve the inequality.

Combining like terms means to group and simplify terms in an expression that have the same variable raised to the same power. For instance, in the expression \(4x - 8 - 4x + 3\), the terms \(4x\) and \(-4x\) are like terms because they both contain the variable \(x\). Similarly, the constants \(-8\) and \(+3\) are like terms.
In our example, we combine these like terms as follows:
\[4x - 4x - 8 + 3 = 0 - 8 + 3 \] Which simplifies to: \[ -5 \]
Combining like terms simplifies the inequality, enabling us to see that \(-5 < 6\) holds true.

Understanding inequality properties is crucial for solving and analyzing inequalities. One key property is that you must reverse the inequality sign when multiplying or dividing by a negative number. However, this rule does not apply here as the solution did not involve multiplying or dividing by a negative number.
An important point in this exercise is recognizing true statements. When simplified, the inequality \(-5 < 6\) is always a true statement. This means any value of \(x\) will satisfy the original inequality.
Properties of inequalities allow us to make these determinations and confirm if solutions cover all possible values through testing simpler, equivalent inequalities.