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

Verified step by step guidance

1. Step 1: Start by isolating the absolute value expression. Divide both sides of the inequality by 3 to simplify: $|2x\; -\; 1|\; \backslash geq\; 7$.
2. Step 2: Recall the definition of absolute value inequalities. For $|A|\; \backslash geq\; B$, the inequality splits into two cases: $A\; \backslash geq\; B$ or $A\; \backslash leq\; -B$. Apply this to the inequality: $2x\; -\; 1\; \backslash geq\; 7$ or $2x\; -\; 1\; \backslash leq\; -7$.
3. Step 3: Solve each inequality separately. For $2x\; -\; 1\; \backslash geq\; 7$, add 1 to both sides to get $2x\; \backslash geq\; 8$, then divide by 2 to find $x\; \backslash geq\; 4$. For $2x\; -\; 1\; \backslash leq\; -7$, add 1 to both sides to get $2x\; \backslash leq\; -6$, then divide by 2 to find $x\; \backslash leq\; -3$.
4. Step 4: Combine the two solution sets. The solution is $x\; \backslash leq\; -3$ or $x\; \backslash geq\; 4$. This represents two disjoint intervals: $(-\backslash infty,\; -3]$ and $[4,\; \backslash infty)$.
5. Step 5: Graph the solution set on a number line. Draw a solid circle at $x\; =\; -3$ and shade to the left to represent $x\; \backslash leq\; -3$. Then, draw a solid circle at $x\; =\; 4$ and shade to the right to represent $x\; \backslash geq\; 4$.

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 that involve expressions within absolute value symbols.

Inequalities

Inequalities express a relationship where one side is not equal to the other, often using symbols like '≥', '≤', '>', or '<'. When solving inequalities, especially those involving absolute values, it is essential to consider both the positive and negative scenarios that satisfy the condition. This dual consideration leads to multiple solution sets that must be graphed appropriately.

Graphing on a Number Line

Graphing on a number line involves representing solutions to inequalities visually. Each solution set is indicated by shading or marking specific regions on the line. For inequalities, it is important to distinguish between open and closed intervals, where closed intervals include the endpoints (indicated by filled circles) and open intervals do not (indicated by open circles). This visual representation helps in understanding the range of solutions.