Problem 77 - A Graphical Approach to Precalculus with Limits

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

We need to solve the equation \(3x - 6 = 0\). Start by adding 6 to both sides to isolate the term with \(x\). \[3x - 6 + 6 = 0 + 6\] This simplifies to \[3x = 6\] Now, divide both sides by 3 to solve for \(x\):\[x = \frac{6}{3} = 2\] Thus, the solution to the equation is \(x = 2\).

Step 2: Solve the Inequality Greater Than Zero

We solve \(3x - 6 > 0\) similarly to the equation. First, add 6 to both sides:\[3x - 6 + 6 > 0 + 6\] Which gives us:\[3x > 6\] Next, divide both sides by 3:\[x > \frac{6}{3}\] Simplify to get:\[x > 2\] In interval notation, this solution is \((2, \infty)\).

Step 3: Solve the Inequality Less Than Zero

For \(3x - 6 < 0\), start by adding 6 to both sides:\[3x - 6 + 6 < 0 + 6\] We then have:\[3x < 6\]Dividing both sides by 3 gives:\[x < \frac{6}{3}\]Simplify to obtain:\[x < 2\]The solution in interval notation is \((-\infty, 2)\).

Key Concepts

Interval Notation

When solving inequalities, interval notation is a useful way to express solutions. It compactly describes where a variable can lie. Let's break down how we use it.

In interval notation:

• Parentheses, \(()\) and \(()\), denote that an endpoint is not included in the interval. For example, \((2, \infty)\) means \(x\) is greater than 2, but not including 2.
• Brackets, \([\) and \(]\), mean endpoints are included. For instance, \([2, 5]\) indicates \(x\) can be between 2 and 5, including both 2 and 5.
• Use "\(\infty\)" or "\(-\infty\)" to indicate that the interval continues indefinitely in either the positive or negative direction. Infinity symbols are always accompanied by parentheses because infinity is never reached.

For our example,

• \((2, \infty)\) describes solutions to \(3x - 6 > 0\), meaning \(x > 2\).
• \((-\infty, 2)\) describes solutions to \(3x - 6 < 0\), meaning \(x < 2\).

By understanding interval notation, you can quickly and efficiently express the range of solutions to inequalities.

Linear Equations

A linear equation is an equation between two variables that gives a straight line when plotted. One variable is often expressed in terms of the other, like in our exercise equation: \(3x - 6 = 0\). Here are the steps to solve linear equations.

1. **Simplify the Equation** - Start by isolating the variable \(x\) by removing any constants from one side. For example, add or subtract values as needed.2. **Clear the Coefficient** - Divide or multiply both sides to solve for \(x\).

In our case:

• Add 6 to both sides: \(3x - 6 + 6 = 0 + 6\), simplifies to \(3x = 6\).
• Divide by 3: \(x = \frac{6}{3} = 2\).

The solution, \(x = 2\), represents the x-value where the line crosses the x-axis. Understanding this gives you the tool to systematically break down similar problems.

Analytical Solution

An analytical solution is a precise, step-by-step solution to a problem using algebraic and logical deductions. This approach leaves no ambiguity about the correct answer. Let’s delve into how we found the solutions in the exercise.

To solve:

• **Equation:** For \(3x - 6 = 0\), rearrange to isolate \(x\). Our solution is the single value \(x = 2\), a straightforward answer.
• **Inequality Greater Than:** For \(3x - 6 > 0\), similar steps lead us to find that all \(x > 2\) satisfy the condition.
• **Inequality Less Than:** For \(3x - 6 < 0\), we find that all \(x < 2\) fit the condition.

The step-by-step analytical approach ensures that no steps are missed, and the logic follows through transparently. Each operation's impact is considered, maintaining accuracy in finding solutions. This builds a strong foundation for tackling more complex algebraic expressions.