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 by 3 to get: $|2x-1|\ge 7$.
2. Recognize that the inequality $|A|\ge B$ implies two separate inequalities: $A\ge B$ or $A\le -B$.
3. Apply this to the expression $2x-1$, resulting in two inequalities: $2x-1\ge 7$ and $2x-1\le -7$.
4. Solve each inequality separately. For $2x-1\ge 7$, add 1 to both sides and then divide by 2 to find $x$. For $2x-1\le -7$, add 1 to both sides and then divide by 2 to find $x$.
5. Graph the solution set on a number line by marking the values of $x$ that satisfy either inequality, using closed circles to indicate that the endpoints are included in the solution set.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities involving expressions that contain it.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). Solving inequalities often involves isolating the variable and considering the direction of the inequality when multiplying or dividing by negative numbers.

Graphing on a Number Line

Graphing on a number line involves representing solutions to inequalities visually. Solutions can be shown as open or closed circles to indicate whether endpoints are included, and shaded regions to represent all numbers that satisfy the inequality. This visual representation helps in understanding the range of solutions and their relationships.