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Solving inequalities is very similar to solving equations, with a few unique steps to remember. Let's take the inequality \(-5 < 4 + x\) as an example. Our primary aim in solving this is to isolate the variable \(x\) on one side. We do this by subtracting \(4\) from both sides of the inequality. This step ensures that we balance the inequality, maintaining equality in terms of arithmetic operations: \(-5 - 4 < 4 + x - 4\). This simplification results in \(-9 < x\). Thus, solving inequalities involves similar procedures to equations, but it is important to solve carefully, especially when multiplying or dividing by a negative number, as it reverses the inequality sign.

In the case of the inequality \(-5 < 4 + x\), no such multiplication or division by a negative number was necessary. Therefore, the direction of the inequality remains the same.

Graphing inequalities involves representing the solution set visually on a number line. For the inequality \(-9 < x\), we need to show that \(x\) can take any value greater than \(-9\). To achieve this, we use a number line—an effective tool for visualizing inequalities.

In this scenario, we start by locating the number \(-9\) on the number line. Since \(-9\) itself is not included in the solution, we draw an open circle over this number. Open circles are used to indicate that the endpoint is not part of the solution, so if the inequality had been \(x > -9\) instead, the open circle would still be used.

Following the open circle, we draw a line extending to the right side of the number line. The direction of the line is crucial as it represents all possible solutions. In this case, extending the line to the right shows that all numbers greater than \(-9\) are valid solutions for the inequality.

A number line is a straightforward way to represent solutions to inequalities visually. By using a number line, we can clearly demonstrate which numbers satisfy the given inequality. In our example, \(-9 < x\), the number line representation starts with an open circle at \(-9\) to show this value is excluded from the solution.

Open and closed circles are crucial elements. An open circle indicates exclusion, whereas a closed circle would denote inclusion. For example, if the inequality were \(x \geq -9\), a closed circle would be used.

After placing the circle at \(-9\), the next step is to indicate all possible solutions by drawing a line that continues to the right. This line serves as a visual cue, highlighting every number that satisfies the inequality. This approach makes understanding and interpreting inequalities much easier, especially when teaching or learning the subject.