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The process of isolating a variable in an inequality involves rearranging the equation so that the variable appears by itself on one side. This method allows us to explicitly solve for the variable's range of values that satisfy the inequality.
To achieve this for

• -4t > 12, divide both sides by -4.
• Since we're dividing by a negative number, remember to flip the inequality sign.

This ensures that the relationship remains true. The outcome is:\[ t < \frac{12}{-4} \] Simplifying this gives: \[ t < -3 \] The solution shows that \( t \) must be less than \(-3\).
Understanding the importance of flipping the sign addresses common mistakes in inequality solutions. Be vigilant with inequality rules, especially when handling negative multiplications or divisions. This critical step helps ensure accurate solutions.

Visualizing inequalities by graphing them can make solutions easier to understand. Start by considering the number line as a visual tool.
For our inequality \( t < -3 \), identify the exact line location that represents \(-3\). Since our inequality is strict (\( t \) less than \(-3\) without equality), we use an open circle at this point. This marks that \(-3\) is not a solution, but boundary.
Next, shade the region of the number line to the left of \(-3\). This shaded region visually represents all possible values of \( t \) that satisfy the inequality. Keeping the graph simple and clear helps reinforce understanding, ensuring accuracy in interpreting inequality solutions.

Representing inequalities on a number line is a foundational skill in understanding their solutions. A number line translates mathematical expressions into a visual format that is easy to comprehend.
The key steps in creating this representation include:

• Identifying important points, such as \(-3\), where the variable begins or ends its set of possible values.
• Using open or closed circles to indicate whether a number is included. Open circles, like in \( t < -3 \), show exclusion.
• Shading the relevant sections of the line to depict the solution set. For \( t < -3 \), shade everything to the left of \(-3\).

Such a visual tool offers direct insight into which values satisfy the inequality immediately, providing a clear picture of the solution's scope.