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An inequality solution is a single value or a range of values that makes an inequality true. In our case, we want to know if inserting the value of 4 into the inequality \(3x - 11 \geq 2\) results in a true statement. Inequalities represent mathematical expressions that indicate one quantity is larger, smaller, or equal under certain conditions compared to another. To determine the solution, we must verify if the inequality holds true when a specific value, like \(x = 4\), is substituted into the equation. This step is critical because it enables understanding of how variable values relate to each other within the bounds set by the inequality.

Algebraic substitution involves replacing a variable in an equation or inequality with a specific number to test its validity or to simplify the expression. In our example, substitution is used to check if \(x = 4\) satisfies the inequality \(3x - 11 \geq 2\). We achieve this by directly inserting the value 4 into the place of \(x\).
This substitution transforms the inequality to \(3(4) - 11\). Follow the arithmetic to simplify this to \(12 - 11\), which equals 1. Now, the original inequality \(3x - 11 \geq 2\) is tested by substituting, and we get 1 on the left side of the inequality. This approach is fundamental in checking solutions and enhancing understanding of how each element within the inequality influences the overall expression.

After substituting, we arrive at the step of inequality comparison. Here, we compare the left side and the right side of the inequality to see if it holds true. After algebraic substitution, our inequality becomes \(1 \geq 2\). In this comparison, 1 is on the left side, and 2 is on the right. Since 1 is not greater than or equal to 2, the inequality does not hold.
This process of comparing sides is essential because it confirms whether a specific value is indeed a solution to the inequality. If the inequality remains valid, that value can be considered a solution. If it doesn't, as in our case where \(4\) is not a solution, the inequality is not fulfilled. This understanding is vital in algebra, as it helps students learn how to check the behavior of values within inequalities.