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Absolute value inequalities, like \( \left| \frac{x-2}{3} \right| < 4 \), tell us how far an expression can be from zero. The absolute value \( |a| \) of any number \( a \) is its distance from zero, regardless of direction on the number line.
In this context, the inequality shows us that the value of \( \frac{x-2}{3} \) must remain within 4 units of zero. This is why we split it into two inequalities: the expression is greater than \(-4\) and less than \(4\). This provides two separate "limits" for \( \frac{x-2}{3} \) to solve for \( x \).

The process:

• If we have an inequality of the form \( |A| < B \), it translates into \( -B < A < B \).
• It creates a range of values that \( A \) can take.
• The absolute value just measures distance from zero, hence we consider both the negative and positive limits.

Interval notation is a concise way of expressing a range of values solutions, especially useful after solving inequalities. In interval notation, we use brackets or parentheses to show which numbers are included or excluded in a set.

• Parentheses \(( )\) are used to indicate that an endpoint is not included in the interval, which is called an open interval.
• Brackets \([ ]\) imply that an endpoint is included, known as a closed interval.

Using our solution \(-10 < x < 14\) from the inequality, it means \( x \) can be any number between \(-10\) and \(14\), but not exactly those numbers.
So, in interval notation, this is written as \((-10, 14)\).

Think of interval notation like a shorthand substitute for describing parts of the number line:

• It allows mathematicians and students to communicate inequalities quickly.
• It is important in expressing solutions clearly and precisely, especially for computer calculations.

Solving inequalities involves finding all possible values of a variable that satisfy the inequality condition. Unlike equations, inequalities often produce ranges or intervals as solutions rather than single values.
Here's a brief guide to solve inequalities like this one:

• Start by manipulating the inequality just like you would an equation. Perform operations to isolate the variable.
• Remember when multiplying or dividing by a negative number, flip the inequality sign.
• For absolute inequalities, split them into two separate inequalities, one for the greater than and one for the less than condition.

In our example, we broke the absolute inequality into two parts and solved them like regular linear inequalities. Then, we found a common solution that satisfies both conditions.
This highlighted the intersection of solutions, giving us the possible range for \( x \).

Understanding how to solve inequalities is key because:

• They describe scenarios where multiple outcomes are possible.
• They form the basis for constraints in optimization and real-world problems.

Always check your solutions by substituting back, ensuring they meet the original inequality conditions.