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Absolute value refers to the distance of a number from zero on the number line, irrespective of direction. It's important because it gives us a way to measure magnitude without worrying about positive or negative signs. When you see the absolute value symbol, represented by two vertical bars \(|\ldots|\), it simply asks for the distance, so \(|-5| = 5\) and \(|5| = 5\). In the original problem, \(|2x + 9| \leq 15\), the expression inside the absolute value can be either positive or negative, as its absolute value has to not exceed 15.
When solving absolute value inequalities, remember the key idea: the scenario covers both when the expression inside is less than or equal to 15 and when it's greater than or equal to -15. This leads to two inequalities, capturing the full range of possible solutions.

Solving inequalities with absolute values involves breaking them down into two separate inequalities. For \(|2x + 9| \leq 15\), we interpret it as two cases:

• \(2x + 9 \leq 15\)
• \(2x + 9 \geq -15\)

By solving each inequality, you'll find the range for the variable. The solution translates into:

• For \(2x + 9 \leq 15\), subtract 9 from both sides, leading to \(2x \leq 6\). Divide by 2 to get \(x \leq 3\).
• For \(2x + 9 \geq -15\), subtract 9 from both sides, which gives \(2x \geq -24\). Divide by 2 to get \(x \geq -12\).

These steps let you find the interval that satisfies both conditions, showing the overlap: \(-12 \leq x \leq 3\). This solution represents all values of \(x\) that meet the original inequality.

Graphing inequalities helps visualize solutions, making them easier to understand. With \(-12 \leq x \leq 3\), you're plotting all x-values that satisfy both parts of the inequality. Begin by drawing a number line.

The endpoint -12 has a closed dot, indicating \(x\) can be exactly -12. Similarly, place a closed dot at 3 for \(x \leq 3\).

Shading the line segment between -12 and 3 shows all the values that satisfy the inequality. These closed intervals mean that every number from -12 to 3, including -12 and 3, is part of the solution.

Graphing this way not only confirms the solution visually but also aids in understanding how inequalities describe ranges of numbers rather than discrete values.