Problem 26 - Elementary Algebra Within Reach

Question

Solve and graph the inequality. $$2 x>3$$

Step-by-Step Solution

Answer

The solution to the inequality \(2x > 3\) is \(x > 1.5\). Graphically, this is represented by an open circle at 1.5 with an arrow extending to the right on the number line.

Step 1: Solve the inequality

In order to solve the inequality \(2x > 3\), the variable \(x\) should be isolated. This can be done by dividing both sides of the inequality by 2, resulting in \(x > 1.5\). Remember that when dividing by a positive number, the direction of the inequality remains the same.

Step 2: Graph the solution

To graph the inequality \(x > 1.5\), a number line is utilized. A circle is drawn at 1.5, and because our inequality sign is 'greater than' (and not 'greater than or equal to'), this circle should be open, indicating that 1.5 is not included in the solution set. Then, an arrow is drawn extending to the right of 1.5, indicating all numbers greater than 1.5 are solutions to the inequality.

Key Concepts

Algebraic Inequalities

An \textbf{algebraic inequality} is a mathematical statement that relates two expressions with an inequality sign, such as <, >, \( \leq \) (less than or equal to), or \( \geq \) (greater than or equal to). The process of solving these inequalities is to find the values of the variable that make the inequality true.

To solve an inequality, similar steps to solving an equation are followed. However, one critical difference in handling inequalities is understanding how they behave under multiplication or division by negative numbers. When multiplying or dividing by a negative number, the inequality direction must be reversed to maintain a true statement. In the example \(2x > 3\), we are looking for all the possible values of \(x\) that make this inequality true. The solution, found through isolating \(x\) by dividing both sides by 2—since 2 is a positive number—does not require flipping the inequality sign, leaving us with \(x > 1.5\).

This solution tells us that any number greater than 1.5 is a solution to the inequality.

Graphing Inequalities

To visually represent the solution to an inequality, \textbf{graphing the inequality} on a number line is extremely helpful. The graph offers a quick way to see all of the possible values that satisfy the inequality.

In the given example \(x > 1.5\), graphing begins with drawing a number line, which represents all real numbers. Then, a crucial decision involves choosing the type of circle that will represent 1.5: an open circle indicates the point 1.5 itself is not included (for 'greater than' or 'less than'), whereas a closed or filled circle indicates that the point is included (for 'greater than or equal to' or 'less than or equal to').

After placing an open circle on 1.5 to reflect the 'greater than' inequality sign, an arrow is drawn extending rightward to show that all numbers greater than 1.5 are part of the solution set. This graphical representation helps users immediately understand the range of values that are solutions to the inequality.

Number Line Representation

The \textbf{number line representation} is a simple but powerful tool to illustrate and understand inequalities. It provides a spatial model of numbers where the position of each number corresponds to its magnitude relative to other numbers.

When you graph an inequality on the number line, you not only show the point at which the inequality might change its value, but also the direction of the values that satisfy the inequality. In the example of \(x > 1.5\), an open circle on 1.5 and an arrow pointing to the right inform us that any point to the right of 1.5 is part of the solution. As an additional tip, when a number line is included in your notes or on a test, be sure to label the line with a scale appropriate for the inequality being graphed, and always indicate with an arrow the direction in which the inequality continues, which signifies that it extends infinitely in that direction.

Furthermore, the number line can accommodate every type of inequality, including compound inequalities by using two arrows or multiple shaded regions. This versatility makes number line representations a central aspect of learning and teaching algebraic inequalities.