Problem 79 - College Algebra

Question

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$3<|2 x-1|$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 2\), and in interval notation, it is \((2, +\infty)\).

Step 1: Rewrite the Absolute Value Inequality

An absolute value of a difference, such as |2x-1| can be rewritten in two parts. Consider both the positive and the negative values that will make the expression true, since absolute value represents distance from zero on a number line. Therefore \(3<|2x-1|\) becomes \(3<2x-1\) and \(-3<2x-1\).

Step 2: Solve Both Inequalities

For \(3<2x-1\), add 1 to both sides to get \(4<2x\) and then divide by 2 to get \(22\). For \(-3<2x-1\), add 1 to both sides to get \(-2<2x\) and then divide by 2 to get \(-12\).

Step 3: Graph the Solution on a Number Line

On a number line, the solution will be represented by an open circle at 2 (since 2 is not included in the solution set) and an arrow pointing to the right indicating all values greater than 2.

Step 4: Write Solution in Interval Notation

In interval notation, the solution will be \((2, +\infty)\). The parentheses indicate that 2 is not included in the solution set.

Key Concepts

Solving Inequalities

Solving inequalities is a fundamental skill in algebra that enables students to find the range of values that satisfy a given mathematical statement. Unlike equations, which have equal values on both sides, inequalities show that one side is either greater than, less than, greater than or equal to, or less than or equal to another.

When it comes to absolute value inequalities, such as the problem at hand, the solution process involves considering two separate scenarios because the absolute value represents a number's distance from zero, not the sign of the number itself. This means for the inequality \(3 < |2x - 1|\), we need to solve for when \(2x - 1\) is greater than 3 and also when \(-(2x - 1)\) is greater than 3, which essentially considers the negative counterpart.

By solving each case, we determine the set of values for \(x\) that make the original inequality true. Remember to reverse the inequality sign when multiplying or dividing by a negative number which hasn't been necessary for this specific problem but is crucial in some cases.

Number Line Graphing

Graphing on a number line is an intuitive way to represent inequalities visually. It helps students clearly see which numbers are included in the solution set. For the inequality solved above, we graph the solution set by marking an open circle on the number line at the value 2 to represent that 2 is not included in the set. This is because the original inequality is strictly 'greater than' and does not include equality.

An open circle is used for inequalities that do not include the endpoint (>, <), while a closed circle is used when the endpoint is included (≥, ≤). To indicate the range of values that satisfy the inequality, an arrow extending to the right or left from the open circle is drawn. In this case, since \(x > 2\), we draw the arrow to the right, demonstrating that all numbers greater than 2 are part of the solution.

Interval Notation

Interval notation is a shorthand method of writing the set of numbers that satisfy an inequality. It is a concise way to express intervals of real numbers, showing the start and end points of the interval along with whether these points are included or excluded from the set.

For instance, the solution to the inequality \(x > 2\) is presented as \((2, +\infty)\) in interval notation. The parenthesis '(' means that 2 is not included (just like the open circle on the number line). Conversely, a bracket '[' would imply the number is included. The symbol \(+\infty)\) indicates that the interval extends indefinitely to the right on the number line, covering all values greater than 2. It's important to not have the infinity symbol enclosed in brackets, as infinity is not a number that can be reached or included in the set.