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Graphing inequalities helps visualize the solutions provided by an inequality. When you solve an inequality, like \[-5<3t+4\leq13\], it represents a range of possible values. Knowing how to graph these on a number line can demystify much of the confusion about what constitutes a solution set.
To graphically show an inequality, you begin by marking important points, such as where the inequality begins and ends.

• Open Circles: These indicate that a number is not included in the solution set. For example, if \(t > -3\), you place an open circle at -3 to show it’s not part of the solution.
• Closed Circles: These indicate that a number is included in the solution set. For \(t \leq 3\), a closed circle at 3 shows it is inclusive.

Draw a line between those key points to depict all the values that satisfy the inequality. This visual aid not only helps in comprehending inequalities but also serves great in checking solutions quickly.

Solving inequalities involves the same foundational steps as solving equations. However, a few pivotal differences are essential to highlight. With inequalities, we aim to find a set of values, not just a single solution.
In our original exercise, we began by splitting the main inequality into two easier-to-manage parts:

• Part 1: \(-5<3t+4\)
• Part 2: \(3t+4\leq13\)

Afterwards, isolate the variable on one side by performing operations like subtraction or division. Additionally, remember an essential rule: if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This step is crucial for preserving the inequality's original meaning. The solutions from both parts inform together where the variable falls on the number line.

Algebra allows us to represent complex problems through symbols and expressions, simplifying the problem-solving processes. In the context of inequalities, algebraic manipulation becomes especially handy.
For instance, to solve an inequality such as \(-5<3t+4\leq13\), we first separate it into two more understandable inequalities. Then we proceed to clear out constants from around the variable:

• Subtract numbers from both sides to alter the inequality without changing its truth.
• Divide or multiply by constants, being mindful of signs since they dictate the inequality direction.

Handling inequalities through algebra not only refines our solving skills but also sharpens logical thinking and problem-solving strategies. This makes algebra a ubiquitous asset in various mathematical contexts.