Problem 82 - Intermediate Algebra

Question

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. \(-\frac{3}{4} x>-\frac{21}{32}\) b. \(-\frac{3}{4} x \leq-\frac{21}{32}\)

Step-by-Step Solution

Answer

a: \((-\infty, \frac{7}{8})\); b: \((-\infty, \frac{7}{8}]\)

Step 1: Solve the inequality in part a

Given the inequality \(-\frac{3}{4} x > -\frac{21}{32}\). To solve for \(x\), we need to isolate it by multiplying both sides by \(-\frac{4}{3}\) (the reciprocal of \(-\frac{3}{4}\)). Remember that multiplying or dividing by a negative number will reverse the inequality sign. Thus, we have:\[ x < \left(-\frac{21}{32}\right) \times \left(-\frac{4}{3}\right) \]Calculating the right side gives:\[ x < \frac{21 \times 4}{32 \times 3} \]Simplifying further:\[ x < \frac{84}{96} \]Divide both the numerator and the denominator by 12 (the greatest common divisor):\[ x < \frac{7}{8} \]

Step 2: Graph the solution set for part a

The solution \(x < \frac{7}{8}\) is represented on a number line by an open circle at \(\frac{7}{8}\) and a shaded line extending to the left. This indicates that the solution does not include \(\frac{7}{8}\) itself, but includes all numbers less than \(\frac{7}{8}\).

Step 3: Write the interval notation for part a

The solution to the inequality \(-\frac{3}{4} x > -\frac{21}{32}\) in interval notation is written as:\((-\infty, \frac{7}{8})\).

Step 4: Analyze part b using part a's solution

Part b is \(-\frac{3}{4} x \leq -\frac{21}{32}\). The inequality sign has changed to 'less than or equal to', so the solution from part a can be used directly with a slight modification to include equality. Thus, the solution for part b is:\(x \leq \frac{7}{8}\).

Step 5: Graph the solution set for part b

For \(x \leq \frac{7}{8}\), we represent this on a number line with a closed circle at \(\frac{7}{8}\) and a shaded line extending to the left. The closed circle indicates that \(\frac{7}{8}\) is included in the solution set.

Step 6: Write the interval notation for part b

The solution to the inequality \(-\frac{3}{4} x \leq -\frac{21}{32}\) in interval notation is:\((-\infty, \frac{7}{8}]\).

Key Concepts

Interval Notation

Interval notation is a concise way of writing subsets of the real number line. It's a powerful tool in mathematics, especially when dealing with inequalities.

When using interval notation, parentheses \( () \) and brackets \( [] \) are used to denote open and closed intervals, respectively. This distinction is crucial as it tells us whether endpoints are included in the interval.

• Open intervals, such as \((-\infty, \frac{7}{8})\), use parentheses to denote that the endpoint is not included (e.g., \(\frac{7}{8}\) is not part of the solution set).
• Closed intervals, like \((-\infty, \frac{7}{8}]\), incorporate brackets to show that the endpoint is included.

By understanding interval notation, students can easily visualize and communicate the range of solutions for inequalities.

Graphing Inequalities

Graphing inequalities on a number line helps make solutions more visually intuitive. This approach serves as a bridge between the abstract algebraic expression and its practical implications.

When graphing:

• Identify critical points from the inequality solution, such as \(\frac{7}{8}\) in our exercise.
• Use open circles to indicate values that are not included in the solution set (e.g., for \(x < \frac{7}{8}\)).
• Employ closed circles for values that are included (e.g., for \(x \leq \frac{7}{8}\)).
• Draw a line or arrow to show the direction of the solution. For example, shading towards negative infinity for solutions less than the critical point.

This graphical method allows students to observe complex solutions in a clear, manageable format, making it easier to interpret inequalities.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to solve for unknown variables. Mastery of this technique is essential for solving inequalities effectively.

In the given exercise, the primary challenge is isolating \(x\) in \(-\frac{3}{4} x > \frac{-21}{32}\). Here's a step-by-step approach:

• Understand the importance of the reciprocal. Since we need to eliminate the coefficient \(-\frac{3}{4}\), multiply by \(-\frac{4}{3}\).
• Remember that multiplying or dividing by a negative reverses the inequality sign. This crucial property changes the inequality from \(>\) to \(<\).
• Simplify the resulting fractions meticulously to obtain the clean inequality form, e.g., \(x < \frac{7}{8}\).

These steps reveal how algebraic manipulation reconciles abstract symbolic math with concrete numeric narratives.

Number Line Representation

A number line representation is an invaluable tool for visualizing solutions to inequalities. It provides a straightforward way to depict the relationship between numbers and solutions.

Here's a quick guide on how to effectively use the number line:

• Mark each significant point with a dot or circle. Open circles signify values not part of the solution (e.g., \(x < \frac{7}{8}\)) while closed circles do the opposite (e.g., \(x \leq \frac{7}{8}\)).
• Draw an arrow or shaded line indicating the range of solutions. This depicts all possible values that satisfy the inequality.

This method transforms numerical solutions into visual concepts, aiding in the comprehensive understanding of inequalities.