Problem 94 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically. Write the solution set in interval notation. Support the answer graphically. $$\frac{2 x+3}{5}-\frac{3 x-1}{2}<\frac{4 x+7}{2}$$

Step-by-Step Solution

Answer

The solution set is \( \left( \frac{-24}{31}, \infty \right) \).

Step 1: Combine the Fractions

Start by finding a common denominator for the fractions on the left side of the inequality. The common denominator of 5 and 2 is 10. Rewrite each term with this common denominator:\[ \frac{2x+3}{5} = \frac{2(2x+3)}{10} = \frac{4x+6}{10} \]\[ \frac{3x-1}{2} = \frac{5(3x-1)}{10} = \frac{15x-5}{10} \]Subtract the second fraction from the first within the inequality:\[ \frac{4x+6}{10} - \frac{15x-5}{10} < \frac{4x+7}{2} \]

Step 2: Simplify the Left Side

Now, subtract the numerators of the fractions on the left side over the common denominator:\[ \frac{4x+6 - (15x-5)}{10} = \frac{4x+6 - 15x + 5}{10} = \frac{-11x+11}{10} \]Thus, the inequality becomes:\[ \frac{-11x+11}{10} < \frac{4x+7}{2} \]

Step 3: Clear the Fractions by Multiplying with LCM

Multiply both sides of the inequality by 10 to eliminate the fractions:\[ -11x + 11 < 5(4x + 7) \]This simplifies the inequality as no fractions remain.

Step 4: Expand and Rearrange Terms

Expand the terms on the right side:\[ -11x + 11 < 20x + 35 \]To isolate the variable, add 11x to both sides:\[ 11 < 31x + 35 \]

Step 5: Solve for x

Subtract 35 from both sides to isolate the term with \( x \):\[ 11 - 35 < 31x \]This simplifies to:\[ -24 < 31x \]Finally, divide both sides by 31:\[ \frac{-24}{31} < x \]

Step 6: Write the Solution in Interval Notation

Convert the solution \( \frac{-24}{31} < x \) into interval notation:The solution is all real numbers greater than \( \frac{-24}{31} \), which is written as:\[ \left( \frac{-24}{31}, \infty \right) \]

Step 7: Graph the Solution

To graph the solution, draw a number line. Mark \( \frac{-24}{31} \) on the line. Since \( x \) is greater than \( \frac{-24}{31} \), use an open circle at \( \frac{-24}{31} \) and shade the line to the right towards infinity to represent all numbers greater than \( \frac{-24}{31} \).

Key Concepts

Analytical Methods

When solving inequalities analytically, the main goal is to isolate the variable using algebraic techniques. This process involves several important steps:

• Combine Fractions: Find a common denominator to consolidate fractions on one side of the inequality.
• Simplify Expressions: Perform operations like addition or subtraction to combine terms.
• Eliminate Fractions: Multiply both sides by the least common multiple to remove fractions and work with simpler expressions.
• Rearrange and Isolate Variable: Use additive and multiplicative operations to shift all terms involving the variable to one side, and constants to the other.
• Solve: Simplify further to extract the inequality's solution in terms of the variable.

Understanding each of these stages is crucial because inaccuracies at any step can lead to incorrect results. A clear understanding of algebraic rules like distribution and balancing equations is pivotal for effective analytical solving.

Interval Notation

Interval notation is a concise way of expressing the set of solutions for inequalities. It uses brackets (or parentheses) to denote the range of possible values.In this exercise, after solving the inequality from -24/31 onwards, the solution set in interval notation is represented as \((\frac{-24}{31}, \infty) \). Here’s how this works:

• Parentheses indicate that the endpoint is not included in the solution set. Thus, \(\frac{-24}{31}\) is not part of the solution.
• Infinity symbol expresses that the set extends beyond any real number to the right, covering all greater numbers.

Interval notation is efficient in conveying the scope of solutions without using verbose descriptions or inequalities. It's important to become familiar with this form to understand mathematical solutions quickly.

Graphical Representation

Visualizing inequalities through graphs provides an intuitive understanding of solutions. Here, the solution is represented on a number line.For the inequality \(x > \frac{-24}{31}\), the graphical representation involves:

• Drawing a number line.
• Marking the point \(\frac{-24}{31}\) with an open circle, which shows that this point is not part of the solution.
• Shading the line to the right of \(\frac{-24}{31}\), indicating that all numbers greater than this value satisfy the inequality.

Graphical representations are particularly useful to visually convey mathematical truths and are crucial in understanding concepts like domain, range, and the nature of solutions to inequalities. By bridging analytical results with visual information, students can better grasp and confirm their understanding of the solution set.