Problem 92 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The solution is \( \left[ \frac{-70}{29}, \infty \right) \).

Step 1: Combine Fractions

First, we need to combine the fractions on the left side of the inequality. The expressions \( \frac{3x}{2} \) and \( \frac{4x}{7} \) have different denominators. To add them, find the least common denominator, which is 14.

Step 2: Rewrite Terms with Common Denominator

Convert each term to a fraction with a common denominator of 14: \( \frac{3x}{2} = \frac{21x}{14} \) and \( \frac{4x}{7} = \frac{8x}{14} \).

Step 3: Add the Fractions

Add the fractions: \( \frac{21x}{14} + \frac{8x}{14} = \frac{29x}{14} \). Now the inequality is \( \frac{29x}{14} \geq -5 \).

Step 4: Eliminate the Fraction

Multiply both sides of the inequality by 14 to eliminate the fraction: \( 29x \geq -70 \).

Step 5: Solve for x

Divide both sides by 29 to solve for \( x \): \( x \geq \frac{-70}{29} \).

Step 6: Write the Solution in Interval Notation

The solution set in interval notation is \( \left[ \frac{-70}{29}, \infty \right) \).

Step 7: Graph the Solution

To graph the solution, draw a number line. Mark the point \( \frac{-70}{29} \), and shade the region to its right, including the point itself, to show all numbers greater than or equal to \( \frac{-70}{29} \).

Key Concepts

Fraction Addition

Adding fractions requires a common denominator. This step ensures that you can combine the numerators effectively. In problems like our exercise, where you're adding fractions like \( \frac{3x}{2} \) and \( \frac{4x}{7} \), it's crucial to tackle this first. Here’s how you proceed:

• Determine the least common denominator (LCD) for the fractions. In this case, it's 14 for the denominators 2 and 7.
• Rewrite each fraction with the LCD. Convert \( \frac{3x}{2} \) into \( \frac{21x}{14} \) by multiplying the numerator and denominator by 7. Convert \( \frac{4x}{7} \) into \( \frac{8x}{14} \) by multiplying the numerator and denominator by 2.
• Add the rewritten fractions: \( \frac{21x}{14} + \frac{8x}{14} = \frac{29x}{14} \).

Adding fractions is like gathering pieces of a puzzle under one frame. Once you have a common denominator, simply add the numerators, while keeping the denominator the same. Be mindful of this method in any problem involving fraction addition.

Interval Notation

Interval notation is a neat way to express the range of solutions for inequalities. After solving the inequality, you need to represent the solution in terms of intervals. Here’s what to keep in mind:

• The notation \( \left[ a, b \right) \) indicates all numbers from \( a \) to \( b \), including \( a \) but not \( b \). The square bracket \([\) implies inclusivity, while the parentheses \(()\) mean exclusivity.
• Our exercise resulted in \( x \geq \frac{-70}{29} \). Hence, in interval notation, it becomes \( \left[ \frac{-70}{29}, \infty \right) \).
• Infinity is always paired with a parenthesis \(()\) as it represents an unbounded limit on one side.

This notation is efficient to use and communicates the solution's boundaries quickly. Remember, precise use of brackets and parentheses matters greatly.

Graphical Representation

Visualizing solutions on a number line helps in understanding the span of solutions. In inequalities, it’s crucial to show clearly what values \( x \) can take.

• First, draw a horizontal number line and locate the point \( \frac{-70}{29} \) on it. This value is approximately \(-2.4138\).
• Since our inequality includes \( x \geq \frac{-70}{29} \), use a solid dot at this point to indicate that the point itself is included in the solution set.
• Shade the number line to the right of \( \frac{-70}{29} \) to show all values greater than this point are part of the solution.

Graphical representation provides a clear and immediate way to grasp which numbers fulfill the inequality conditions. It serves as a bridge between analytical solutions and visual understanding.