Absolute value inequalities involve expressions with absolute values that are set in inequality form. Solving these requires understanding how absolute values work and breaking down the inequality into different cases. Here’s a basic guide:

1. Basic Principle of Absolute Value Inequalities

For an expression $|x| \leq a$ or $|x| \geq a$, we handle it as follows:

1. If $|x| \leq a$:
   This means $-a \leq x \leq a$.
   Example: $|x| \leq 5$ implies $-5 \leq x \leq 5$.

2. If $|x| \geq a$:
   This means $x \leq -a$ or $x \geq a$.
   Example: $|x| \geq 3$ implies $x \leq -3$ or $x \geq 3$.

2. Solving Steps for Simple Absolute Value Inequalities

1. Isolate the absolute value expression if it’s not alone.
2. Determine if the inequality is of the form $\leq$ (and type) or $\geq$ (or type).
3. Write the compound inequality according to the rules above.
4. Solve each part of the inequality separately.

Examples:

1. Example 1: Solve $|x + 2| < 4$
   Since $|x + 2| < 4$ (and type inequality): $-4 < x + 2 < 4$ Solve by isolating $x$: $-4 - 2 < x < 4 - 2$ $-6 < x < 2$ Solution: $-6 < x < 2$.

2. Example 2: Solve $|3x - 1| \geq 5$
   Since $|3x - 1| \geq 5$ (or type inequality): $3x - 1 \leq -5 \quad \text{or} \quad 3x - 1 \geq 5$ Solve each inequality:

   • For $3x - 1 \leq -5$: $3x \leq -4$ $x \leq -\frac{4}{3}$
   • For $3x - 1 \geq 5$: $3x \geq 6$ $x \geq 2$ Solution: $x \leq -\frac{4}{3}$ or $x \geq 2$.

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Related Questions

1. How do we handle inequalities when the inequality sign flips during the solving process?
2. How can we solve absolute value inequalities with quadratic expressions inside?
3. What are practical applications of absolute value inequalities?
4. How do compound inequalities differ from absolute value inequalities?
5. What happens if an absolute value inequality has no solution?

Tip:

Always remember to interpret absolute value inequalities as either "and" or "or" types, depending on whether they are $\leq$ or $\geq$ inequalities.