Problem 66 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -12<6(x-3) \leq 0 $$

Step-by-Step Solution

Answer

\((1, 3]\)

Step 1: Distribute the 6

The inequality given is \(-12 < 6(x-3) \leq 0\). First, we distribute the 6 across the expression \(x-3\), which gives us:\(-12 < 6x - 18 \leq 0\).

Step 2: Add 18 to all parts

Next, to isolate the term with \(x\), we add 18 to each part of the inequality. This yields:\(-12 + 18 < 6x \leq 0 + 18\).Simplifying, we have:\(6 < 6x \leq 18\).

Step 3: Divide each part by 6

To solve for \(x\), divide each part of the inequality by 6:\(\frac{6}{6} < x \leq \frac{18}{6}\).Simplify the fractions to get:\(1 < x \leq 3\).

Step 4: Graph the solution set

On a number line, the solution set \(1 < x \leq 3\) is represented by an open circle at 1 and a closed circle at 3, with a line connecting the two points. This indicates that \(x\) can be any number greater than 1 and up to and including 3.

Step 5: Write solution in interval notation

The interval notation for the solution set \(1 < x \leq 3\) is written as:\((1, 3]\).

Key Concepts

Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. It's compact and visually clear, which makes it ideal for describing the solutions of inequalities. When you use interval notation, you will often see parentheses or brackets:

• Parentheses \( ( \, \text{or} \, ) \) indicate that the endpoint is not included in the set. This is like an open circle on a graph.
• Brackets \( [ \, \text{or} \, ] \) mean that the endpoint is part of the solution. This is how a closed circle is shown on a number line.

Let's look at our solution \(1 < x \leq 3\): - The endpoint \(1\) is not included, so we use an open parenthesis: \( (1\).- The endpoint \(3\) is included, so we use a bracket: \(3]\). Thus, the interval notation becomes \( (1, 3] \). This tells us clearly that \(x\) is greater than \(1\), but less than or equal to \(3\).

Graphing Solutions

Graphing the solutions of an inequality helps visualize the range of possible values easily. You use a number line to illustrate which values of \(x\) meet the conditions. For example, in our situation with \(1 < x \leq 3\), we can represent this visually on the number line.
When graphing:

• Use an open circle to show a value that is not part of the solution. For \(x = 1\), you would place an open circle, indicating that 1 is not included.
• Use a closed circle for a value that is part of the solution. Here, \(x = 3\) would be depicted by a filled-in circle.
• Draw a line connecting these circles to show that all numbers between them are solutions to the inequality.

This graphical representation allows you to immediately see which values are included in the solution set. It is especially helpful for quick verifications.

Algebraic Expressions

Algebraic expressions form the core of solving inequalities, as they allow us to manipulate equations to find the solution set. An algebraic expression can include numbers, variables, and operations like addition and multiplication. The goal is to isolate the variable to determine its value range.

For this particular inequality, we started with \( -12 < 6(x-3) \leq 0 \). By distributing the 6, we simplified it to \( -12 < 6x - 18 \leq 0 \). Here are some crucial steps:

• Distributing: Multiply the 6 across the expression inside the parentheses, making \(6x - 18\).
• Balancing: Add 18 across all parts to keep the inequality balanced.
• Dividing: To solve for \(x\), divide each part by 6.

By manipulating the expression logically, we found \(1 < x \leq 3\), which is the range of our variable that satisfies the original inequality. Understanding these steps helps in solving similar algebraic problems.