Problem 81 - 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) \(x+12=4 x\) (b) \(x+12>4 x\) (c) \(x+12<4 x\)

Step-by-Step Solution

Answer

(a) \(x = 4\); (b) \((-\infty, 4)\); (c) \((4, \infty)\).

Step 1: Solve the Equation for (a)

First, we will solve the equation \(x + 12 = 4x\). Start by moving all terms involving \(x\) to one side. Subtract \(x\) from both sides to get:\[ 12 = 4x - x \]Next, simplify the equation:\[ 12 = 3x \]Finally, divide both sides by 3 to isolate \(x\):\[ x = \frac{12}{3} = 4 \]The solution to (a) is \(x = 4\).

Step 2: Solve the Inequality for (b)

Next, we solve the inequality \(x + 12 > 4x\). Start by moving all terms involving \(x\) to one side. Subtract \(x\) from both sides:\[ 12 > 4x - x \]Simplify the right side:\[ 12 > 3x \]Now, divide both sides by 3 to solve for \(x\):\[ \frac{12}{3} > x \]Simplify the expression:\[ 4 > x \] or \(x < 4\)In interval notation, the solution is \((-\infty, 4)\).

Step 3: Solve the Inequality for (c)

Finally, solve the inequality \(x + 12 < 4x\). Start by moving all terms involving \(x\) to one side. Subtract \(x\) from both sides:\[ 12 < 4x - x \]Simplify the right side:\[ 12 < 3x \]Now, divide both sides by 3 to solve for \(x\):\[ \frac{12}{3} < x \]Simplify the expression:\[ 4 < x \] or \(x > 4\)In interval notation, the solution is \((4, \infty)\).

Key Concepts

Analytical Solution

The term **analytical solution** involves solving equations and inequalities in a step-by-step manner, using algebraic techniques. This approach focuses on rearranging and simplifying algebraic expressions to isolate the variable. Let's explore this with two examples, one being an equation and the other an inequality.

For the equation \( x + 12 = 4x \), the goal is to find the value of \( x \) that makes the equation true. Begin by organizing terms so that one side of the equation holds only \( x \). This requires the subtraction of \( x \) from both sides:
- Move \( x \) to the other side to get: \( 12 = 4x - x \).

By simplifying the equation, you consolidate the variable on one side and constants on the other. Thus:
- Simplify: \( 12 = 3x \).
- Divide by 3 to isolate \( x \): \( x = 4 \).

To solve an inequality like \( x + 12 > 4x \), rearrange in a similar manner. Move terms involving \( x \) together:
- Subtract \( x \) from both sides: \( 12 > 3x \).

Once simplified, divide by 3 to isolate \( x \), giving \( 4 > x \), or equivalently, \( x < 4 \). Analytical solutions reveal not just a single answer but also an approach applicable to a variety of similar problems.

Interval Notation

**Interval notation** provides a concise way of representing the solution set of an inequality. It is a standardized form that clearly indicates which numbers are included in a set resulting from an inequality solution.

For an inequality such as \( x < 4 \), the solution describes all numbers less than 4. In interval notation, you write this as \((-\infty, 4)\).
- Parentheses \(()\) indicate that the boundary number (in this case 4) is not included.
- \(-\infty\) and \(\infty\) represent indefinite continuation in the negative and positive directions, respectively.

Similarly, if you have \( x > 4 \), the interval notation becomes \((4, \infty)\), signifying numbers greater than 4, excluding 4 itself. Understanding how to use interval notation allows you to communicate complex solution sets neatly and efficiently.

Simplifying Expressions

The process of **simplifying expressions** involves reducing expressions to their simplest form, making them easier to work with or understand. This requires combining like terms, reducing coefficients, and performing arithmetic operations.

For instance, take the equation \( x + 12 = 4x \). To simplify, subtract \( x \) from both sides:
- This leads to expression \( 12 = 4x - x \).
- Combine like terms to get \( 12 = 3x \).

For inequalities like \( x + 12 < 4x \), similar steps apply:
- Subtract \( x \) from both sides to simplify the inequality to \( 12 < 3x \).
- Finally, divide by 3, simplifying to \( 4 < x \).

These simplified expressions are more direct and make it easier to find solutions. Mastery in simplifying expressions underpins effective problem-solving in algebra, allowing you to focus on the heart of the problem without unnecessary complexity.