Problem 64 - College Algebra with Modeling and Visualization

Question

Solve the compound linear inequality graphically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth whenever appropriate. $$ 0.2 x<\frac{2 x-5}{3}<8 $$

Step-by-Step Solution

Answer

The solution set is \((3.57, 14.5)\).

Step 1: Understand the compound inequality

The given compound inequality is \( 0.2x < \frac{2x - 5}{3} < 8 \). This means that the expression \( \frac{2x - 5}{3} \) is greater than \( 0.2x \) and less than 8.

Step 2: Solve the first part of the inequality

Start with the inequality \( 0.2x < \frac{2x - 5}{3} \). To eliminate the fraction, multiply every term by 3: \[ 3(0.2x) < 2x - 5 \] which simplifies to \[ 0.6x < 2x - 5 \].Rearrange to solve for \( x \): \[ 0.6x - 2x < -5 \]\[ -1.4x < -5 \]Divide by -1.4, and remember to reverse the inequality sign: \[ x > \frac{5}{1.4} \approx 3.57 \].

Step 3: Solve the second part of the inequality

Now, solve the inequality \( \frac{2x - 5}{3} < 8 \). Multiply each term by 3 to clear the fraction:\[ 2x - 5 < 24 \].Add 5 to both sides:\[ 2x < 29 \].Divide both sides by 2:\[ x < 14.5 \].

Step 4: Combine the solution

We now have two conditions: \( x > 3.57 \) and \( x < 14.5 \). Combining these, the solution set is \[ 3.57 < x < 14.5 \].

Step 5: Express the solution in interval notation

The interval notation for the solution set is \((3.57, 14.5)\). These are rounded to the nearest tenth, which matches the solution we found.

Key Concepts

Graphical Representation

When working with compound inequalities like \( 0.2x < \frac{2x - 5}{3} < 8 \), a graphical approach can be immensely helpful. Visualizing the problem allows you to see where the solutions of individual inequalities intersect.

To begin, let's graph the solutions of each part separately. For the inequality \( 0.2x < \frac{2x - 5}{3} \), after solving, we find that \( x > 3.57 \). This indicates that all points to the right of \( 3.57 \) on the number line are solutions. Next, consider \( \frac{2x - 5}{3} < 8 \), which simplifies to \( x < 14.5 \). This tells us that all points to the left of \( 14.5 \) are solutions.

When you graph these two inequalities on a number line:

• Place an open circle at \( 3.57 \) to show that \( 3.57 \) itself is not included, and shade to the right.
• Place another open circle at \( 14.5 \) and shade to the left of it.

Where these two shaded areas intersect is your solution set, \( (3.57, 14.5) \). This region represents all \( x \) values that satisfy both parts of the compound inequality. Graphical representation helps in visually verifying your calculations.

Interval Notation

Writing solutions in interval notation is a concise way to express the solution set of an inequality. It captures the start and end of the interval where the inequality holds true. For our example, where we solved the compound inequality, the solution was \( 3.57 < x < 14.5 \).

In interval notation, this is written as \( (3.57, 14.5) \). Here’s why:

• The round parentheses "(" and ")" indicate that the endpoints are not included. These points are known as exclusive boundaries.
• If endpoints were included, square brackets "[" and "]" would be used instead.

With \( (3.57, 14.5) \), the parentheses show that \( x \) can be any value between \( 3.57 \) and \( 14.5 \), but not exactly \( 3.57 \) or \( 14.5 \) themselves. Interval notation provides an efficient way to convey information clearly and is a standard part of mathematical communication.

Solving Inequalities

Solving inequalities involves finding all values of a variable that makes the inequality true. In this exercise, we dealt with a compound inequality, \( 0.2x < \frac{2x - 5}{3} < 8 \), which presented dual conditions that the expression \( \frac{2x - 5}{3} \) had to satisfy simultaneously.

For compound inequalities, the strategy is to break them into separate parts. First solve each part individually:

• Start with \( 0.2x < \frac{2x - 5}{3} \). Clear the fraction by multiplying through by 3 and solve for \( x \) to find \( x > 3.57 \).
• Then, handle \( \frac{2x - 5}{3} < 8 \), again clearing the fraction by multiplication, yielding \( x < 14.5 \).

Next, combine these results. Both conditions must hold, leading to the combined solution \( 3.57 < x < 14.5 \).

It's important to remember:

• If you multiply or divide by a negative number when solving, reverse the inequality sign.
• Check your solution by substituting a value within the interval to ensure it satisfies the original inequality.

Each step in solving inequalities is crucial for ensuring the accuracy of your solution.