Problem 81 - 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) \(x+12=4 x\) (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: Isolate x (Equation)

Starting with the equation \(x + 12 = 4x\), we need to isolate \(x\). Begin by subtracting \(x\) from both sides: \(12 = 3x\).

Step 2: Solve for x (Equation)

To solve for \(x\), divide both sides by 3: \(x = 4\). Thus, the solution to the equation is \(x = 4\).

Step 3: Set Up Inequality and Isolate x (Greater Than)

For the inequality \(x + 12 > 4x\), subtract \(x\) from both sides: \(12 > 3x\).

Step 4: Solve the Inequality (Greater Than)

Divide both sides by 3 to solve for \(x\): \(x < 4\). The solution in interval notation is \((-\infty, 4)\).

Step 5: Set Up Inequality and Isolate x (Less Than)

For the inequality \(x + 12 < 4x\), subtract \(x\) from both sides: \(12 < 3x\).

Step 6: Solve the Inequality (Less Than)

Divide both sides by 3 to solve for \(x\): \(x > 4\). The solution in interval notation is \((4, \infty)\).

Key Concepts

Interval Notation

Interval notation is a shorthand way of expressing a range of numbers. It's especially useful in mathematics when describing the solution sets of inequalities or intervals on a number line.

Here's how interval notation works:

• Parentheses ")" and "(": These indicate that an endpoint is not included in the interval, which is called "open" interval.
• Brackets "]" and "[": These indicate that an endpoint is included, which is called a "closed" interval.

For example, when we write \((-\infty, 4)\), it means "all numbers less than 4," but not including 4 itself. Here, \(-\infty\) means the range is extending indefinitely in the negative direction. Interval notation simplifies complex sets so that you can quickly comprehend which numbers are included in or excluded from the solution set.

Isolating Variables

Isolating variables is a crucial skill in algebra used to solve equations and inequalities. The goal is to get the variable on one side and the numbers on the other, effectively "isolating" the variable. This allows us to solve for the variable's value.

Here's how to isolate variables:

• Step 1: Look at the equation or inequality. Identify which term contains the variable you want to solve for.
• Step 2: Use basic arithmetic operations (such as adding, subtracting, multiplying, or dividing) to rearrange the equation. Eliminate other terms while keeping the balance.
• Example: In solving \(x + 12 = 4x\), subtract \(x\) from both sides to get \(12 = 3x\). Then divide by 3 to isolate \(x\). You’ll find \(x = 4\).

Effective isolation involves understanding the operations' inverse to undo them, thus achieving a straightforward outcome like \(x = a\) or \(x > a\).

Inequalities

Inequalities are mathematical expressions involving the symbols \(<, \, >, \, \leq, \, \geq\) which indicate that one side is less than, greater than or equal to the other. Solving inequalities is similar to solving equations, but with a twist.

• Step 1: Start by isolating the variable, just like you would with equations.
• Step 2: Pay attention to the direction of the inequality sign. If you multiply or divide by a negative number, the inequality sign flips direction.

• Greater Than Example: To solve \(x + 12 > 4x\), subtract \(x\) from both sides to get \(12 > 3x\), then divide by 3: \(x < 4\). This is read as "x is less than 4" and its solution is noted in interval notation as \((-\infty, 4)\).
• Less Than Example: With \(x + 12 < 4x\), after subtracting \(x\) and dividing by 3, you find \(x > 4\), or "x is greater than 4," written as \((4, \infty)\).

Remember, inequalities help describe not just specific numbers, but whole ranges that satisfy a particular condition.