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When it comes to understanding inequalities, one of the most useful visual tools is graphing them on a number line. Graphing is not just about plotting points, but also about representing all the possible solutions to an inequality. For example, if we have an inequality like \( t \text{≤} 11 \), we'd first locate the number 11 on the number line. Then, to indicate that \( t \) can be any number less than or equal to 11, we shade the entire line to the left of 11. This shaded area signals all the numbers that satisfy the inequality. Graphing on a number line helps to provide a clear visual of the range of acceptable values, making it easier to understand and interpret inequalities.

To solve an inequality, it is essential to isolate the variable on one side. This process is similar to solving a typical equation, but you're working with inequalities like \(\geq\), \(\leq\), \(>\), or \(<\), instead of just \(=\). The main goal is to get the variable by itself in order to clearly see which values make the inequality true. To achieve this, you perform various algebraic operations such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. In our example, subtracting 57 from both sides simplified the inequality to \( -4t \text{≥} -44 \), setting the stage for us to further isolate \( t \) by dividing by -4.

It's crucial to remember that when you divide or multiply both sides of an inequality by a negative number, the direction of the inequality sign flips. This rule is vital because it helps maintain the true relationship between the two sides. For instance, if we have \( -4t \text{≥} -44 \) and we divide both sides by -4 to isolate \( t \), the \(\text{≥}\) sign becomes \(\text{≤}\), resulting in \( t \text{≤} 11 \). Without this important step, the final answer would be incorrect. This property is unique to inequalities and highlights one of the key differences between inequalities and equations.

A number line in the context of inequalities is a tool that allows us to communicate which numbers satisfy the inequality. One might wonder how to represent the variable with multiple, even infinite solutions. This is where the number line becomes handy. After isolating the variable and finding the critical value (such as \( t = 11 \)), we use an open or closed circle on the number line to denote whether the critical value is included (\(\leq\) or \(\geq\)) or not (\(<\) or \(>\)). For \( t \text{≤} 11 \), we'd draw a closed circle at 11 because 11 is part of the solution set. Then, we shade the line to the left to signify all the values that \( t \) can take. The inclusion of a graphic representation with the number line solidifies understanding by visually demonstrating the range of possibilities.