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When solving compound inequalities, interval notation offers a concise way to express the solution set. It captures in one simple expression the range of values that satisfy the inequality. For the inequality \[-9 \leq x + 8 < 1\], after breaking it down into two parts, we found\[-17 \leq x\] and \[x < -7\]. These solutions need to be joined to show the range of potential values for \(x\). In interval notation, this combined solution is written as:

• \([-17, -7)\)

This notation means that the set includes all \(x\) values starting from \(-17\) and going up to, but not including, \(-7\). The square bracket, \([\), at \(-17\) indicates that \(-17\) is part of the set, while the round parenthesis, \()\), at \(-7\) indicates that \(-7\) is not part of the set.
This method of writing is efficient for expressing ranges on number lines, especially when compared to listing all numbers one by one.

Graphing inequalities on a number line is an excellent visual method to represent the solution of an inequality. For our compound inequality, the solution \([-17, -7)\) transforms into a shaded region on the number line. Here's how you do it:
Begin by marking \(-17\) and \(-7\) on the number line. Then, between these marks, draw a line or shade the region to indicate all numbers in this range satisfy the inequality.

• A closed dot on \(-17\) shows that \(-17\) itself is included in the solution. This is due to the \(\leq\) sign in the initial inequality.
• An open dot on \(-7\) signifies that \(-7\) is not part of the solution because the inequality is strict (\(<\)).

Graphing offers a clear and immediate way to interpret compound inequalities. It helps students see connections between numeric ranges and their algebraic representations, making concepts concrete through visualization.

To solve a compound inequality such as\[-9 \leq x + 8 < 1 \], follow a straightforward method by approaching each part of the inequality separately.

First, address the inequality \(-9 \leq x + 8\) by doing the following:

• Subtract 8 from both sides to isolate \(x\), which gives you \(-17 \leq x\). This tells us that \(x\) must be greater than or equal to \(-17\).

Next, solve the second inequality, \(x + 8 < 1\):

• Similarly, subtract 8 from both sides, resulting in \(x < -7\). This indicates \(x\) must be less than \(-7\).

Bringing both parts together, you get the combined solution: \(-17 \leq x < -7\). By isolating \(x\), and accurately manipulating mathematical expressions, algebraic solutions systematically simplify inequalities into more digestible parts, laying the foundation for correct interpretation and graphing of the results.