Solve each inequality. Give the solution set in interval notation. . | 3x - 4 | ≥ 2

Verified step by step guidance

1. Start by understanding that the inequality $|3x-4|\ge 2$ involves an absolute value, which means we need to consider two separate cases: one where the expression inside the absolute value is greater than or equal to 2, and another where it is less than or equal to -2.
2. For the first case, solve the inequality without the absolute value: $3x-4\ge 2$. Add 4 to both sides to isolate the term with $x$: $3x\ge 6$.
3. Divide both sides by 3 to solve for $x$: $x\ge 2$. This gives us one part of the solution set.
4. For the second case, solve the inequality $3x-4\le -2$. Again, add 4 to both sides: $3x\le 2$.
5. Divide both sides by 3 to solve for $x$: $x\le \frac{2}{3}$. Combine the solutions from both cases to express the solution set in interval notation.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that measure the distance of a number from zero on the number line. The inequality |A| ≥ B means that A is either greater than or equal to B or less than or equal to -B. Understanding how to break down absolute value inequalities into two separate cases is crucial for finding the solution set.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b', which is essential for expressing solution sets of inequalities clearly.

Solving Inequalities

Solving inequalities involves finding the values of a variable that satisfy the given inequality. This process may include isolating the variable, applying properties of inequalities, and considering the direction of the inequality sign. It is important to check the solution by substituting back into the original inequality to ensure correctness.