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Inequality solving involves finding the values of a variable that make an inequality true. In the provided exercise, our task is to determine whether a specific value for \( x \) (here, \( x = -2 \)) makes a given inequality true. The inequality given is \( |x + 2| < 4 \). To solve this, we begin by substituting the value of \( x \) and simplifying the expression. The absolute value reflects the distance from zero, meaning \( |a| \, \) corresponds to \( a \) if \( a \geq 0 \) and \( -a \) if \( a < 0 \). With the substitution of \( x = -2 \), it becomes \( |-2 + 2| < 4 \), which simplifies to \( |0| < 4 \). Since 0 is indeed less than 4, the inequality is satisfied.

• Substituting and simplifying are key steps in verifying inequalities.
• Remember that absolute values depict distance from zero, always non-negative.

Breaking down each step aids in understanding the behavior of inequalities and verifying solutions.

The substitution method is a powerful technique used to check if a specific value satisfies an equation or inequality. In this problem, we substitute \( x = -2 \) into the inequality \( |x + 2| < 4 \).The objective is straightforward:

• Replace \( x \) with the given number.
• Perform any necessary arithmetic operations.
• Simplify to evaluate the validity of the original statement.

For the inequality \( |x + 2| < 4 \), substituting \( x = -2 \) results in the absolute value expression \( |0| \). This becomes \( 0 < 4 \). By evaluating the simplified expression, we can ascertain whether the inequality is true. It's evident here that substitution acts as a test, confirming if specific numbers satisfy given conditions.

Precalculus is laden with concepts that provide the groundwork for calculus. Among these are absolute values and inequalities, both integral to understanding relationships between numbers and expressions. Absolute value models distance, indicating how far a number is from zero on the number line. In relation to inequalities, it helps define ranges that solutions can occupy. Precalculus often uses inequalities to describe this kind of relationship, making it crucial to grasp these preliminary approaches. Consider these essentials:

• Absolute Values: Always non-negative, representing distance.
• Inequalities: They describe wider sets of solutions compared to equations, highlighting the concept of range and conditionality in solutions.
• Real-World Application: These concepts model everyday problems such as determining tolerances in engineering or economics.

Understanding these concepts ensures a smooth transition to more advanced topics, establishing a base upon which calculus heavily relies.