Problem 77 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(3 x-6=0\) (b) \(3 x-6>0\) (c) \(3 x-6<0\)

Step-by-Step Solution

Answer

(a) \(x = 2\); (b) \((2, \infty)\); (c) \((-\infty, 2)\).

Step 1: Solve the Equation

For the equation \(3x - 6 = 0\), add 6 to both sides to get \(3x = 6\). Then, divide both sides by 3 to isolate \(x\), obtaining \(x = 2\).

Step 2: Solve the Inequality (Greater Than)

For the inequality \(3x - 6 > 0\), add 6 to both sides to get \(3x > 6\). Then, divide both sides by 3 to solve for \(x\), which gives \(x > 2\). The solution in interval notation is \((2, \infty)\).

Step 3: Solve the Inequality (Less Than)

For the inequality \(3x - 6 < 0\), add 6 to both sides to get \(3x < 6\). Then, divide both sides by 3 to solve for \(x\), which gives \(x < 2\). The solution in interval notation is \((-\infty, 2)\).

Key Concepts

Understanding Inequalities

Inequalities are expressions that determine the relative size or order of two values. Unlike equations, which show equality, inequalities reveal if a value is less than, greater than, equal to or not equal to another value. In mathematics, these are represented by symbols:

• < for less than
• > for greater than
• ≤ for less than or equal to
• ≥ for greater than or equal to

When solving inequalities, it's important to remember that the direction of the inequality sign can change under certain conditions. For instance, when both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign is reversed. This is a vital rule to keep in mind, especially when dealing with complex inequalities.
Inequalities are often solved in a manner similar to equations, with addition, subtraction, multiplication, or division used to isolate the variable on one side of the inequality. From there, we can determine the range of values that satisfy the inequality.

Interval Notation

Interval notation is a mathematical method used to describe the set of solutions of an inequality. It provides a concise way to express the range of numbers that form part of the solution. The notation uses brackets and parentheses:

• Parentheses, ( or ), indicate that an endpoint is not included in the interval.
• Brackets, [ or ], signify that an endpoint is included in the interval.

For example, the inequality solution from our problem, which states that \(x > 2\), is represented in interval notation as \((2, \infty)\). This means that \(x\) can be any number greater than 2, but not 2 itself.
Similarly, \(x < 2\) would be expressed as \((-\infty, 2)\), indicating \(x\) is less than 2, but not equal to 2.
Infinity symbols (\(\infty\) or \(-\infty\)) are always paired with parentheses because infinity is not a number that can be reached or included. Understanding how to represent solutions in interval notation is crucial for communicating mathematical solutions effectively and efficiently.

Solving Equations

Solving equations is a foundational skill in mathematics, often involving finding the value of a variable that satisfies the equation. When solving linear equations, like the one provided in the exercise, the goal is to isolate the variable on one side of the equation. Let's break down the example:
Given the equation \(3x - 6 = 0\),

• First, add 6 to both sides to eliminate the constant term from one side: \(3x = 6\).
• Then, divide both sides by 3 to solve for \(x\): \(x = 2\).

By performing the same operations on both sides of the equation, we maintain the balance and accurately solve for the unknown variable. This approach is systematic and can be applied to a wide range of linear equations.
Equations can be more complex, involving multiple steps, fractions, or larger expressions, but the principle of performing inverse operations to isolate the variable remains the same. Understanding and practicing these methods develops not only proficiency but also confidence in tackling various mathematical problems.