Problem 55 - College Algebra with Modeling and Visualization

Question

Solve the linear inequality graphically. Write the solution set in set-builder notation. Approximate endpoints to the nearest hundredth whenever appropriate. $$ 5 x-4>10 $$

Step-by-Step Solution

Answer

The solution is \( \{ x \mid x > 2.8 \} \).

Step 1: Solve the Inequality Algebraically

First, we need to solve the inequality \(5x - 4 > 10\). Add 4 to both sides to get \(5x > 14\). Then, divide both sides by 5 to solve for \(x\). This gives us \(x > \frac{14}{5}\), which simplifies to \(x > 2.8\).

Step 2: Graph the Inequality on a Number Line

To graph \(x > 2.8\), plot an open circle at \(x = 2.8\) to indicate that this value is not included in the solution. Since the inequality is greater than, draw a line extending to the right from the open circle, down the number line, which represents all numbers greater than 2.8.

Step 3: Write the Solution Set in Set-Builder Notation

The solution set can be expressed in set-builder notation as \(\{ x \mid x > 2.8 \}\). This notation indicates the set of all \(x\) values that satisfy the inequality.

Key Concepts

Graphical Solutions

Graphical solutions for linear inequalities provide a visual way to understand the solution set, enhancing comprehension. To solve an inequality graphically, you begin by identifying its solution on the number line. For the given inequality $5x - 4 > 10$, the algebraic solution was found to be $x > 2.8$. Next, you represent this solution visually. Here’s how you can do it step by step:

• Identify the endpoint: In this case, the endpoint is 2.8. Since the inequality symbol is ' >', 2.8 is not included in the solution set.
• Use an open circle: Place an open circle around 2.8 on the number line. This shows that 2.8 itself is not part of the solution.
• Shade the line: Because the inequality is 'greater than', shade the part of the number line that is to the right of 2.8. This shaded region represents all values of $x$ that satisfy the inequality.

Graphical solutions provide an intuitive way to see how a solution behaves across values.

Set-Builder Notation

Set-builder notation is a concise way to express the solution set of an inequality. It explains which values meet the criteria of the inequality without listing them all. For the inequality \(x > 2.8\), the set-builder notation is written as \( \{ x \mid x > 2.8 \} \). This notation can be broken down like this:

• Curly braces \(\{ \} \): These enclose the entire set.
• The variable \(x\): Indicates what we are considering as part of the solution set.
• The vertical bar \(\mid\): Reads as "such that" or "for which".
• The condition \(x > 2.8\): Describes the values that satisfy the inequality, essentially defining the elements of the set.

This form of notation provides a precise mathematical expression for individuals familiar with the language of sets, offering clarity at a glance.

Number Line

The number line is a valuable tool for graphically representing real numbers and for understanding the nature of solutions to inequalities. When dealing with an inequality like $x > 2.8$, the number line allows you to:

• Visualize quantity: It displays numbers in order from smallest to largest, with equal intervals, making it a superb tool for grasping relations and distances between numbers.
• Represent solutions: You use symbols on the number line to indicate which numbers are part of the solution.
  • Open circles show that a boundary value is not included in the solution.
  • Shaded areas or arrows indicate the direction in which the inequality is satisfied (to the right for ">" or to the left for "<").
• Apply corrections quickly: Easy to use for checking and comparing solutions.

Number lines are a straightforward, visual way of interpreting and analyzing inequalities, useful in both solution checking and presentation.