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To solve an inequality like \(|5 - 2x| > 7\), we start by removing the absolute value signs. The property \(|A| > B \text{ or } -B\) remains true for any B > 0. This means we should split our inequality into two separate parts: \((5 - 2x > 7) \text{ or } (5 - 2x < -7)\).

Next, solve each part separately. For \((5 - 2x > 7)\), subtract 5 from both sides to get \(-2x > 2\). Divide by -2 and remember to flip the inequality sign to get \((x < -1)\).

For the second part, \((5 - 2x < -7)\), subtract 5 from both sides to get \(-2x < -12\). Divide by -2 and flip the sign to get \((x > 6)\).

Solve the combined solution: \((x < -1\text{ or } x > 6)\).

This final result represents the values that satisfy the original inequality.

Instead of writing solutions in a long form, we use interval notation to make things simpler and more concise.

For our inequality result \((x < -1\text{ or } x > 6)\), the interval notation is written as:

• For \(x < -1\): \((-\infty, -1)\)
• For \(x > 6\): \((6, \infty)\)

We use round brackets to represent intervals where endpoints are not included (this is called an open interval).

The union (denoted by \(\cup\)) combines them to show the full solution in interval notation: \((-\infty, -1) \cup (6, \infty)\).

Visualizing inequalities helps understand the solution better. For \((x < -1\text{ or } x > 6)\) graph:

Draw a number line and locate points -1 and 6. Use open circles around these points to show they are not included in the intervals.

Shade the line to the left of -1 to indicate values less than -1. Similarly, shade the line to the right of 6 for values greater than 6. This graphical representation clearly shows all possible solutions.

Absolute values measure the distance from zero on a number line, so they are always positive or zero. When solving inequalities, important properties are:

• \(|A| > B\), where \(B > 0\), translates to \(A > B \text{ or } A < -B\).
• \(|A| < B\) leads to \(-B < A < B\).

These properties help break down equations into simpler, non-absolute inequalities we can solve more easily.

Understanding absolute values and their properties ensures accurate solutions and simplifies complex expressions.