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Linear inequalities are similar to linear equations but instead of an equality sign, they use inequality signs like \(>\), \(<\), \(\geq\), or \(\leq\). Solving them involves finding the range of values for which the inequality holds true.
A key aspect of linear inequalities is that operations such as adding, subtracting, multiplying, or dividing a number from both sides of the inequality keep it balanced.
However, be careful if you multiply or divide by a negative number, as this will flip the inequality sign.
In our example, we began with the inequality \(9x + 4 > 4x - 1\) and through simplification, we found that \(5x > -5\). Lastly, we solved for \(x\) by dividing by 5, resulting in \(x > -1\). This means that \(x\) can be any number greater than -1.

Once we solve the inequality, we can visualize the solution by graphing it. Graphing solutions helps us quickly see which values satisfy the inequality.
In our solved inequality \(x > -1\), we need to represent all values greater than -1 on a number line or graph.
To do this:

• Identify the critical point, which is -1 in this case.
• Since \(x > -1\), values that satisfy the inequality are not equal to -1, so we use an open circle to represent -1.
• To show that all numbers greater than -1 are solutions, draw a line that extends to the right of the open circle, encompassing all greater numbers.

This graphical representation makes it clear which regions of the number line correspond to solutions of the inequality.

The number line is a simple yet powerful tool to represent the solutions of inequalities. It allows for a clear visual interpretation of the range of values involved.
By using a number line, one can easily understand and communicate which numbers are part of the solution.
For the inequality \(x > -1\):

• Place an open circle at -1 to indicate that -1 itself is not included in the solutions.
• Extend a line to the right from the open circle, showing that all greater numbers are valid solutions.
• This format clearly demonstrates that every point to the right of -1 is a solution without ambiguity.

In conclusion, representing solutions on a number line is an intuitive way to show which values satisfy the inequality, aiding in better comprehension and communication.