Solve each group of equations and inequalities analytically....

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Key Concepts

Absolute Value

The absolute value of a number or expression represents its distance from zero on the real number line, disregarding its sign. This concept is fundamental in understanding equations and inequalities that involve absolute values, as it lays the groundwork for interpreting expressions like |A| = B, where B is non-negative.

Absolute Value Equations

When solving equations of the form |expression| = constant, the key technique is to consider the definition of absolute value, which leads to two separate linear equations: one where the expression itself equals the positive constant and another where it equals the negative constant. This bifurcation is essential for finding all possible solutions of the equation.

Absolute Value Inequalities

Solving inequalities that involve absolute values requires interpreting the inequality in terms of distance. For an inequality of the form |expression| ? constant, the solution is expressed as a compound inequality, meaning the expression lies within a fixed distance from zero. Conversely, for |expression| ? constant, the solution is typically the union of two ranges, representing values that are at least the specified distance away from zero. Understanding these translations is crucial for correctly solving and graphing such inequalities.

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