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Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols like \(\leq\), \(<\), \(>\), or \(\geq\). They represent a range of values rather than an exact number. In the given problem, we have the inequality \(8 + x \leq -9\). To solve it, we want to find the values of \(x\) that make this statement true.

The goal here is to isolate \(x\) on one side of the inequality. By performing operations like addition, subtraction, multiplication, or division, we can simplify the inequality. It is important to note that if we multiply or divide both sides of an inequality by a negative number, we must flip the inequality symbol. This step is crucial in maintaining the true relationship between the values.

A number line is a visual tool that helps us graphically represent solutions to inequalities. It is a straight line where each point corresponds to a number. On this line, positive numbers are to the right of zero, and negative numbers are to the left.

In plotting solutions for inequalities, a number line provides clarity on which numbers satisfy the inequality. For example, if \(x \leq -17\), we would mark the number \(-17\) on the number line and shade to the left to show all numbers less than or equal to \(-17\) are solutions. A number line is intuitive and makes it easier to understand the extent of the solution set.

Graphically representing solutions to inequalities helps in visualizing which values satisfy the inequality. In the solution \(x \leq -17\), the number line is used for this graphical representation. Here's how you can do it:

• Locate \(-17\) on the number line.
• Draw a filled circle at \(-17\) to indicate that \(-17\) itself is included in the solution (because of the \(\leq\) sign).
• Shade a line to the left of \(-17\) to cover all numbers less than \(-17\) as these are also solutions.

This method clearly shows the solution set in a way that is both visual and easy to understand. The filled circle and shading are key components of illustrating inequality solutions on a number line.

The solution to an inequality tells us the set of possible values for a variable. In \(x \leq -17\), this means any number \(x\) that is less than or equal to \(-17\) will make the inequality true.

Unlike equations, which have a specific value, inequalities have a range of values, and this range is called the solution set. To express these solutions, you can use:

• Set notation, for example, \(\{ x \mid x \leq -17 \}\).
• Interval notation, which looks like \((-\infty, -17]\).

Understanding and being able to express solutions in different notations provide a more in-depth comprehension of inequality solutions, allowing for flexibility in presentation and interpretation.