Problem 34 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$-\frac{1}{2} \leq \frac{4-3 x}{5} \leq \frac{1}{4}$$

Step-by-Step Solution

Answer

The solution is \([\frac{11}{12}, \frac{13}{6}]\).

Step 1: Break the Compound Inequality

The given compound inequality is \[-\frac{1}{2} \leq \frac{4-3x}{5} \leq \frac{1}{4}\].We will break it into two separate inequalities:1. \(-\frac{1}{2} \leq \frac{4-3x}{5}\)2. \(\frac{4-3x}{5} \leq \frac{1}{4}\).

Step 2: Solve the First Inequality

Solve \(-\frac{1}{2} \leq \frac{4-3x}{5}\):1. Multiply both sides by 5 to clear the fraction: \(-\frac{1}{2} \times 5 \leq 4-3x\) \ \(-\frac{5}{2} \leq 4-3x\).2. Subtract 4 from both sides: \(-\frac{5}{2} - 4 \leq -3x\) \ \(-\frac{13}{2} \leq -3x\).3. Divide both sides by -3 and flip the inequality sign: \(\frac{13}{6} \geq x\).4. Rewrite: \(x \leq \frac{13}{6}\).

Step 3: Solve the Second Inequality

Solve \(\frac{4-3x}{5} \leq \frac{1}{4}\):1. Multiply both sides by 5 to clear the fraction: \(4-3x \leq \frac{5}{4}\).2. Subtract 4 from both sides: \(-3x \leq \frac{5}{4} - 4\) \ \(-3x \leq -\frac{11}{4}\).3. Divide both sides by -3 and flip the inequality sign: \(x \geq \frac{11}{12}\).

Step 4: Find the Solution Set

Combine the solutions from Steps 2 and 3:\( \frac{11}{12} \leq x \leq \frac{13}{6} \).This is the solution to the compound inequality.

Step 5: Express in Interval Notation

The solution set \( \frac{11}{12} \leq x \leq \frac{13}{6} \) in interval notation is \([\frac{11}{12}, \frac{13}{6}]\).

Step 6: Graph the Solution Set

Draw a number line. Mark the points \( \frac{11}{12} \) and \( \frac{13}{6} \). Shade the region between \( \frac{11}{12} \) and \( \frac{13}{6} \), including both endpoints, as they are included in the interval with square brackets.

Key Concepts

Compound Inequalities

Compound inequalities are an essential concept in mathematics. They involve two separate inequalities that are combined into one statement, usually using the words "and" or "or." In our exercise, we have a compound inequality \(-\frac{1}{2} \leq \frac{4-3x}{5} \leq \frac{1}{4}\).
It's crucial to break it down into two individual inequalities before solving. This gives us:

• \(-\frac{1}{2} \leq \frac{4-3x}{5}\)
• \(\frac{4-3x}{5} \leq \frac{1}{4}\)

Handling them separately makes the problem easier and allows you to focus on solving each part correctly. Once each inequality is solved, their solutions are combined, keeping the nature of compound inequalities in mind. They represent a range where both conditions are satisfied.

Interval Notation

Interval notation is a simple way to describe sets of numbers on a number line. It's highly useful when expressing solutions to inequalities. In our exercise, the solution to the compound inequality \(\frac{11}{12} \leq x \leq \frac{13}{6}\)
is expressed in interval notation as \([\frac{11}{12}, \frac{13}{6}]\).
Here’s how it works:

• Brackets \([\underline{\phantom{xxx}} ]\) are used to include numbers (closed interval), meaning the endpoints are part of the solution.
• Parentheses \((\underline{\phantom{xxx}} ]\) would be used if the endpoint was not included (open interval), which is not the case here.

This concise format is excellent for quickly understanding solution ranges. It's essential to pay attention to whether endpoints are included or not, as that affects how you interpret the inequality's solution.

Inequality Solving

Solving inequalities is much like solving equations, but with some important twists.
The goal is to isolate the variable on one side while maintaining balance. The crucial difference is the rule about reversing the inequality sign if you multiply or divide both sides by a negative number. Here's a brief summary of the steps:

• Clear fractions by multiplying every term by the lowest common denominator, as seen with the multiplication by 5 in our exercise.
• Simplify the inequality by adding or subtracting terms on both sides, just like when moving constants or variables in an equation.
• When you multiply or divide by a negative number, remember to flip the inequality sign. This step can often be tricky but is crucial for the correct solution.

The solution to each inequality must be carefully checked also to ensure it fits into the overall problem when dealing with compound inequalities.

Graphing Solutions

Graphing is a great way to visualize the solution set of an inequality. For our exercise, once the inequality is solved, we can depict its solution on a number line. Here is how to approach:

• Identify the endpoints \(\frac{11}{12}\) and \(\frac{13}{6}\) on a number line and mark them. Ensure clarity in representation, especially with mixed fractions.
• Shade the region between these points to show that any number in this interval is part of the solution.
• Use solid dots at \(\frac{11}{12}\) and \(\frac{13}{6}\) because both are included due to the closed interval notation.

Graphing gives a visual understanding of where the solutions lie and is a useful skill to check the correctness of the solved inequalities. This method supports significantly in confirming the range derived through compound inequality solving.