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Solving linear inequalities is quite similar to solving linear equations. The primary goal is to isolate the variable on one side of the inequality. Let's take our example: \( t+1 < 6 \). A good first step is to simplify the inequality by eliminating any constants from the side of the inequality that includes the variable.

To do this, subtract 1 from both sides to maintain the balance. Doing so, we have \( t+1 - 1 < 6 - 1 \), which simplifies to \( t < 5 \).

Our inequality \( t < 5 \) suggests that any value of \( t \) that is less than 5 is a solution. Remember, when solving inequalities, if you multiply or divide by a negative number, the inequality sign flips direction. However, that's not necessary for this simple inequality.

Graphing inequalities provides a visual representation of possible solutions on a number line. For instance, for the inequality \( t < 5 \), we need to consider all values of \( t \) that are less than 5.

When we graph it, we start by identifying the critical point on the number line. Here, it's the number 5. Since 5 is not part of the solution (since our inequality shows "less than 5" and not "less than or equal to 5"), we use an **open circle** at 5. An open circle signifies that the number itself is not included in the set of possible solutions.

Next, draw a line extending left from the open circle, indicating that all numbers less than 5 satisfy the inequality. By graphing, we create an intuitive way to view solutions of inequalities.

The number line is a powerful tool for representing solutions to inequalities. It helps to visualize not just individual solutions, but an entire set of solutions at a glance.

To effectively represent \( t < 5 \) on a number line:

• Draw a horizontal line which will serve as the number line.
• Mark appropriate intervals covering a range, like from -10 to 10, for good measure.
• Locate and mark the number 5. Place an open circle on this point because 5 itself is not included.
• Draw a line extending to the left from the open circle, showing that all numbers less than 5 are included in the solution set.

By using these steps, anyone can read the graph and understand what values satisfy the inequality.