In Exercises 59–94, solve each absolute value inequality. - 4|1 - x| < - 16

Verified step by step guidance

1. Recognize that the inequality \(-4|1 - x| < -16\) can be simplified by dividing both sides by \(-4\).
2. When dividing or multiplying an inequality by a negative number, remember to reverse the inequality sign. This gives us \(|1 - x| > 4\).
3. The absolute value inequality \(|1 - x| > 4\) can be split into two separate inequalities: \(1 - x > 4\) and \(1 - x < -4\).
4. Solve the first inequality \(1 - x > 4\) by subtracting 1 from both sides to get \(-x > 3\), then multiply by \(-1\) and reverse the inequality to get \(x < -3\).
5. Solve the second inequality \(1 - x < -4\) by subtracting 1 from both sides to get \(-x < -5\), then multiply by \(-1\) and reverse the inequality to get \(x > 5\).

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving inequalities that involve expressions within these bars.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal, using symbols like <, >, ≤, or ≥. In the context of absolute value inequalities, they help determine the range of values that satisfy the condition. For instance, solving |x| < a involves finding values of x that lie within a certain distance from zero.

Properties of Inequalities

When manipulating inequalities, certain properties must be observed, such as the fact that multiplying or dividing by a negative number reverses the inequality sign. This is particularly important when isolating variables in absolute value inequalities. Understanding these properties ensures accurate solutions and helps avoid common mistakes in solving inequalities.