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The distributive property is a fundamental component in solving linear inequalities and equations. It allows us to simplify expressions by distributing a factor across terms inside parentheses. Consider the expression given in the exercise: \(3(2x + 1)\). By applying the distributive property, we multiply each term inside the parentheses by 3. This process can be broken down into two steps:

• Multiply \(3\) by \(2x\) to get \(6x\).
• Multiply \(3\) by \(1\) to get \(3\).

So, \(3(2x + 1)\) simplifies to \(6x + 3\). Using the distributive property helps to simplify expressions, making it easier to solve the inequality.

Once you have applied the distributive property, the next step is to simplify the expressions by combining like terms. Like terms are terms within an expression that have the same variables raised to the same power. For example, in the inequality:\[8x + 3 > 6x + 3 + x + 5\]The terms \(6x\) and \(x\) are like terms because they both have the variable \(x\). You can add these terms together:

• \(6x + x = 7x\)

Adding the constant terms \(3\) and \(5\) gives you \(8\). Now the expression simplifies to:\[8x + 3 > 7x + 8\]Combining like terms is essential as it reduces the complex expression into simpler terms, which is a crucial step before isolating variables.

Isolating the variable is a critical step in solving inequalities or equations. To find the solution for \(x\), we need to get \(x\) by itself on one side of the inequality. Starting from\[8x + 3 > 7x + 8\]Proceed by eliminating \(7x\) from the right side, which we do by subtracting \(7x\) from both sides:

• \(8x - 7x + 3 > 8\)

This simplifies to:\[x + 3 > 8\]The next step is to isolate \(x\) by removing the constant term on the left side. Subtract 3 from both sides:

• \(x + 3 - 3 > 8 - 3\)

Which leads to:\[x > 5\]Isolating variables requires performing operations that "undo" any addition, subtraction, multiplication, or division affecting the variable. This process allows us to solve for the variable’s value in the inequality or equation.