Problem 13 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ \frac{2}{3} x-\frac{1}{2}<0 $$

Step-by-Step Solution

Answer

The solution set is \( x < \frac{3}{4} \).

Step 1 - Isolate the variable term

First, add \(\frac{1}{2}\) to both sides of the inequality \( \frac{2}{3} x - \frac{1}{2} < 0 \). This gives: \( \frac{2}{3} x < \frac{1}{2} \)

Step 2 - Clear the fraction

To eliminate the fraction, multiply both sides of the inequality by \( \frac{3}{2} \) to get: \( x < \frac{3}{4} \)

Step 3 - Illustrate the solution on the number line

Draw a number line and mark the point \( \frac{3}{4} \). Since \( x < \frac{3}{4} \), shade the region to the left of \( \frac{3}{4} \) and use an open circle at \( \frac{3}{4} \) to indicate that \( \frac{3}{4} \) is not included in the solution set.

Key Concepts

Variable Isolation

When solving inequalities, one of the most important steps is to isolate the variable. This means getting the variable (like \(x\)) by itself on one side of the inequality sign. In our example, \(\frac{2}{3} x - \frac{1}{2} < 0\), we need to remove the fractional constants. To do this, add \(\frac{1}{2}\) to both sides. This results in \(\frac{2}{3} x < \frac{1}{2}\). Now, \(x\) is isolated with a coefficient of \(\frac{2}{3}\) on one side of the inequality. This makes the problem more straightforward.

Fraction Elimination

Next, we tackle the fraction. Fractions can complicate things, so we aim to eliminate them to simplify our equation. In the inequality \(\frac{2}{3} x < \frac{1}{2}\), the variable \(x\) is being multiplied by \(\frac{2}{3}\). To eliminate this fraction, multiply both sides of the inequality by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This step gives us \( x < \frac{3}{4}\). Now, the inequality is much easier to work with since \(x\) is now isolated and there are no fractions left.

Number Line Illustration

To visualize the solution set of the inequality \( x < \frac{3}{4} \), we can use a number line. A number line helps us see where the solutions lie. Here’s how to do it:

• Draw a horizontal line and label it as the number line.
• Mark the point \(\frac{3}{4}\) on this line.
• Since the inequality is \( x < \frac{3}{4} \), we shade the region to the left of \(\frac{3}{4}\).
• Use an open circle at \(\frac{3}{4}\) to show that \(\frac{3}{4}\) is not included in the solution set.

This visual representation helps to clearly understand the range of values satisfying the inequality.