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Inequality solutions involve finding all possible values of a variable that make an inequality true. Unlike equations, which specify exact values, inequalities can cover a range or set of values. However, in some special cases, like the inequality \( |A| \leq 0 \), they can pinpoint a specific value. For an inequality involving absolute values, the solution often requires setting the expression inside the absolute value equal to zero.

Here is why: The absolute value expression \(|A|\) is always zero or positive. So, when asked to solve \(|A| \leq 0\), the only possibility is that \(A\) itself must be zero. Solving for \(x\) then becomes the focus of the exercise.

• If the expression within the absolute value results in zero, then the inequality holds only for that specific value of the variable.
• For more common inequalities, such as \(|A| < b\) or \(|A| > b\), the solution set provides a range of possible values.

Understanding these steps helps identify whether the solution is a range or a specific value.

Absolute value equations are concerned with expressions within absolute value bars. An absolute value denotes the distance of a number from zero on the number line, always resulting in a non-negative value.

When dealing with equations like \( \left|\frac{5}{2} x + 2\right| = 0 \), it's important to realize that the absolute value being zero occurs only when its contents are zero. Thus, the expression \(\frac{5}{2} x + 2\) equals zero.

• To solve, set the equation inside the absolute value equal to zero and solve for \(x\).
• Subtract, add, or multiply to isolate \(x\).

By understanding this principle, students can more easily tackle absolute value equations, applying the same strategy across a range of problems.

Graphing inequalities involves visually representing the solution set on a number line. For the equation \( \left|\frac{5}{2} x + 2\right| \leq 0 \), we determined that \(x = -\frac{4}{5}\) is the only solution.

In this case, the graph on the number line is just a single point. Here's how graphing would proceed:

• First, identify the solution from solving the inequality or equation.
• Place a closed dot on the number line at \(-\frac{4}{5}\) to denote the solution's inclusion.
• A closed dot signifies that the endpoint itself is part of the solution set.
• No shading is required here since the solution is not a range.

Visual representations assist in confirming and understanding solution sets' positional context, simplifying interpretation of outcomes.