Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 3-5 x<11 $$

Step-by-Step Solution

Answer

The solution is \((-\frac{8}{5}, \infty)\).

Step 1: Subtract 3 from both sides

The inequality given is \( 3 - 5x < 11 \). To isolate the variable term, subtract 3 from both sides of the inequality: \( 3 - 5x - 3 < 11 - 3 \). Simplifying both sides gives us: \( -5x < 8 \).

Step 2: Divide both sides by -5

To solve for \( x \), divide both sides of the inequality by \( -5 \). Remember that dividing by a negative numbers reverses the inequality sign:\( \frac{-5x}{-5} > \frac{8}{-5} \).Simplifying, this results in: \( x > -\frac{8}{5} \).

Step 3: Express the solution in interval notation

The solution \( x > -\frac{8}{5} \) can be expressed in interval notation as \( \left(-\frac{8}{5}, \infty\right) \). This means that the solution includes all numbers greater than \(-\frac{8}{5}\).

Key Concepts

Interval Notation

Interval notation is a concise way of representing sets of numbers, typically solutions to inequalities. It describes the range of values that satisfy the inequality. In this system, we use parentheses or brackets to denote intervals. Parentheses indicate that an endpoint is not included, while brackets indicate that an endpoint is included. For instance:

• Parentheses - \((a, b)\): All numbers between \(a\) and \(b\), but not including \(a\) or \(b\).
• Brackets - \([a, b]\): All numbers between and including \(a\) and \(b\).
• Mixed notation - \((a, b]\): All numbers between \(a\) without including it, but including \(b\).

To express the solution \(x > -\frac{8}{5}\) in interval notation, we write \((-\frac{8}{5}, \infty)\), signifying all numbers greater than \(-\frac{8}{5}\) and extending to infinity.

Solving Inequalities

Solving inequalities is akin to solving equations but with a few extra rules due to the inequality sign. The goal is to manipulate the inequality to isolate the variable, while keeping track of how transformations affect the inequality:

• Addition or Subtraction: You may add or subtract the same value from both sides of an inequality without changing the inequality sign. For example, from \(3 - 5x < 11\), we subtract 3 from both sides, resulting in \(-5x < 8\).
• Multiplication or Division: When multiplying or dividing both sides of an inequality by a positive number, the inequality sign stays the same. However, multiplying or dividing by a negative number requires flipping the inequality sign. In our problem, dividing by \(-5\) reverses the sign, turning \(-5x < 8\) into \(x > -\frac{8}{5}\).

It's crucial to solve inequalities step-by-step, ensuring that each transformation respects these rules, to maintain the integrity of the inequality.

Inequality Properties

Understanding the basic properties of inequalities helps solve them correctly. Inequalities express a range of values instead of a single value, which provides flexibility in solutions but also demands careful handling:

• Transitive Property: If \(a < b\) and \(b < c\), then \(a < c\). This property helps in chaining several inequalities together.
• Addition Property: If \(a < b\), then \(a + c < b + c\). This allows consistent increases across an inequality.
• Multiplication Property: If \(a < b\) and \(c > 0\), then \(ac < bc\). If \(c < 0\), multiplying both sides by \(c\) reverses the inequality, turning \(ac > bc\).

These properties are fundamental when solving inequalities and applying operations such as addition, subtraction, multiplication, and division. They help ensure the solution remains correct under various transformations.