Problem 23 - Precalculus Mathematics for Calculus

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$\frac{1}{3} x+2<\frac{1}{6} x-1$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -18)\).

Step 1: Eliminate Fractions

First, eliminate the fractions by multiplying all terms by the least common denominator (LCD) of the fractions involved. For the fractions \( \frac{1}{3} \) and \( \frac{1}{6} \), the LCD is 6. Multiply each term by 6 to get: \[ 6 \left(\frac{1}{3}x\right) + 6(2) < 6 \left(\frac{1}{6}x\right) - 6(1) \]Simplifying, this gives:\[ 2x + 12 < x - 6 \]

Step 2: Isolate the Variable

Next, get all the \(x\) terms on one side and constant terms on the other side. Subtract \(x\) from both sides:\[ 2x - x + 12 < -6 \]Simplifying, this gives:\[ x + 12 < -6 \]

Step 3: Solve for x

Now, isolate \(x\) by subtracting 12 from both sides:\[ x + 12 - 12 < -6 - 12 \]Simplifying, this gives:\[ x < -18 \]

Step 4: Express Solution in Interval Notation

The inequality \( x < -18 \) is expressed in interval notation as \( (-\infty, -18) \). This represents all values less than -18.

Key Concepts

Interval Notation

Interval notation provides a concise way to describe a set of numbers, especially useful in inequalities. It uses parentheses and brackets to indicate the range of values included in the solution set. An open interval, denoted with parentheses, signifies that the endpoints are not included in the set. Conversely, a closed interval uses brackets, indicating that the endpoints are included.

For instance, in the solution to our inequality problem, the inequality \( x < -18 \) translates to interval notation as \((-\infty, -18)\). This tells us that \( x \) can be any number less than -18. In this interval, \(-\infty\) is always accompanied by an open parentheses because infinity represents a concept, not a number, hence it cannot be included in the set.

Always remember when using interval notation:

• Use \((-\infty\) or \(\infty\)) with parentheses because they are not actual numbers.
• Use \([\text{number}]\) to include a number; use \((\text{number})\) to exclude it.

Graphing Inequalities

Graphing inequalities allows us to visualize the solution set on a number line. This visual representation can make it easier to understand the range of possible values for the variable in question.

To graph an inequality like \( x < -18 \), you need to draw a number line and represent the inequality as follows:

• Locate the number -18 on the number line.
• Use an open circle at -18, indicating that this value is not included in the solution set.
• Draw a line extending to the left from -18 to represent all the numbers less than -18.

This clear and visual method helps solidify your understanding of what the inequality represents. Remember, an open circle is used because the inequality is strict (\(<\) or \(>\)), meaning -18 itself is not part of the solution.

Solving Inequalities

Solving linear inequalities follows similar steps to algebraic equations, with a few additional rules to ensure correct solutions. When solving the inequality \( \frac{1}{3} x+2<\frac{1}{6} x-1 \), we begin by eliminating fractions. This is achieved by multiplying all terms by the least common denominator (LCD), which simplifies the inequality considerably.

Next, isolate the variable on one side of the inequality. This involves moving terms containing the variable to one side, and constants to the other. After simplification, the inequality might appear as \( x+12<-6 \). Subtracting 12 from both sides isolates \( x \), resulting in \( x<-18 \).

An essential rule to remember when solving inequalities is that if you multiply or divide both sides by a negative number, you must flip the inequality sign. This ensures the inequality remains true, preserving the correct solution.