Solve and graph the solution set on a number line: 3|2x-1| ≥ 21

Verified step by step guidance

1. Start by isolating the absolute value expression. Divide both sides of the inequality \(3|2x - 1| \geq 21\) by 3 to get \(|2x - 1| \geq 7\).
2. Recall that for an absolute value inequality \(|A| \geq B\) (where \(B > 0\)), the solution splits into two cases: \(A \geq B\) or \(A \leq -B\).
3. Apply this to \(|2x - 1| \geq 7\), which gives two inequalities: \(2x - 1 \geq 7\) and \(2x - 1 \leq -7\).
4. Solve each inequality separately: For \(2x - 1 \geq 7\), add 1 to both sides and then divide by 2. For \(2x - 1 \leq -7\), add 1 to both sides and then divide by 2.
5. Express the solution set as the union of the two intervals found, and then graph these intervals on a number line, using closed circles to indicate that the endpoints are included.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve them, consider the definition of absolute value as distance from zero, leading to two cases: one where the expression inside is greater than or equal to the positive value, and one where it is less than or equal to the negative value.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves dividing or subtracting terms to simplify the inequality, making it easier to apply the definition of absolute value and split into separate inequalities.

Graphing Solution Sets on a Number Line

Graphing the solution set involves representing all values of the variable that satisfy the inequality on a number line. Use open or closed circles to indicate whether endpoints are included, and shade the regions that represent the solution intervals, helping visualize the range of possible values.