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The concept of absolute value relates to the distance of a number from zero on the number line, without considering direction. This means it is always non-negative.
In mathematical terms, the absolute value of a number \( x \) is denoted as \( |x| \) and can be defined as:

• \( |x| = x \) if \( x \geq 0 \)
• \( |x| = -x \) if \( x < 0 \)

The absolute value essentially "strips" away any negative sign. This property is especially helpful when comparing values in inequalities because it allows us to consider only the magnitude, ignoring the sign.
Consider the inequality \( \left| \frac{x}{x+1} \right| < 1 \). Solving it involves understanding that the expression inside the absolute value must stay within the range of -1 and 1. This gives us two simultaneous inequalities:

• \( -1 < \frac{x}{x+1} < 1 \)

By breaking it down into simpler parts, solving absolute value inequalities becomes more manageable.

Graphing inequalities involves representing the solution set of an inequality on a number line or a coordinate plane. It helps visualize which part of the number line or plane satisfies the inequality. For inequalities involving absolute values, such as \( \left| \frac{x}{x+1} \right| < 1 \), the process includes plotting the expressions on a graph and looking for regions.
Here’s a simple way to graph:

• Identify the function and the constant on the other side of the inequality. Here, we graph \( y = \left| \frac{x}{x+1} \right| \) and the line \( y = 1 \).
• The inequality shows we want the function's value to remain below 1, so we find where the graph dips below the line \( y = 1 \).
• Mark the points where the graph intersects \( y = 1 \). These are critical as they can signal where the inequality changes.
• Shade the region satisfying the inequality, considering the domain restrictions.

In this example, graphing shows the solution is where the function paints below the line, which in practical terms means \( -\frac{1}{2} < x < \infty \) while excluding \( x = -1 \).
Graphing thus aids in confirming algebraic solutions and understanding where inequalities hold true.

The domain of a function is the set of all possible input values (\( x \)-values) for which the function is defined. It's crucial to determine the domain because it tells us where the function can actually operate.
For the expression \( \frac{x}{x+1} \), the key issue is the denominator:\
It can't be zero because division by zero is undefined. This gives the constraint \( x eq -1 \) because when \( x = -1 \), the denominator \( x+1 \) becomes zero.

Here's how to determine the domain for these functions:

• Look for values where the denominator or any square root becomes zero or negative.
• Exclude these critical values from the domain.

With the inequality \( \left| \frac{x}{x+1} \right| < 1 \), the domain is all real numbers except \( x = -1 \).
Understanding the domain simplifies the solution process and prevents errors related to undefined values during the solution of an inequality.