Solve each absolute value inequality. - 4|1 - x| < - 16

Verified step by step guidance

1. Step 1: Start by isolating the absolute value expression. Divide both sides of the inequality by -4. Remember, dividing by a negative number reverses the inequality sign. The inequality becomes: $|1\; -\; x|\; >\; 4$.
2. Step 2: Recall the definition of absolute value inequalities. For $|A|\; >\; B$, the inequality splits into two cases: $A\; >\; B$ or . Apply this to the inequality: $1\; -\; x\; >\; 4$ or .
3. Step 3: Solve each case separately. For the first case, $1\; -\; x\; >\; 4$, subtract 1 from both sides to isolate $-x$: $-x\; >\; 3$. Then divide by -1 (reversing the inequality sign): .
4. Step 4: For the second case, , subtract 1 from both sides to isolate $-x$: . Then divide by -1 (reversing the inequality sign): $x\; >\; 5$.
5. Step 5: Combine the solutions from both cases. The solution to the inequality is or $x\; >\; 5$. This represents the values of $x$ that satisfy the original inequality.

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 where one side is not equal to the other, using symbols like <, >, ≤, or ≥. In the context of absolute value inequalities, they indicate the range of values that satisfy the condition. For instance, solving |x| < a means finding all x values that are within a distance a 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 essential when isolating variables in absolute value inequalities. Understanding these properties helps ensure that the solutions derived from the inequalities are valid.